Organization: Pearson Education Product Name: Envision Integrated Mathematics II Product Version: v1.0 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-8d545551-f0f9-38f0-a8fc-1f9da322c683 Timestamp: Monday, January 14, 2019 03:53 PM EST Status: WARNINGS Conformant: true ***** WARNINGS ***** Resource Validation Results The document is valid but contains some warnings. Warnings (2017) imsmanifest.xml: Failed to locate resource 'I_709655a8-f4b8-33df-93e7-24b1fc9b9ad9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d6880efb-db3a-37c6-8cdc-db08accda90c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_51943444-6ed0-3c1d-b669-40243093514f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49279c8c-fed3-3f76-a543-eaf4bc5763c2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90588d58-b3f6-3ffc-a77d-a1c93716b3cc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fbb7b36e-988f-3d6c-93a4-cd5df6b64f0a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91a1fc9d-ce35-3da2-973c-5ed26ad2d067_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7a761553-f517-3bb0-b585-a74508f6404a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b03ad8e3-4c69-309f-a960-9926abf605c1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e201df7f-1184-3f7e-9e6f-454afbf33afa_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0e05be72-f01c-308f-87c0-9c56a6465876_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16286976-c51d-382e-919b-2e37452dcf67_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a62d84b-a483-3fcb-8e75-9389b184deb2_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_690eeb39-6876-33a0-aadb-24447c506207_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2f20babb-7efc-3d87-b7f9-bd4021596e4f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e201df7f-1184-3f7e-9e6f-454afbf33afa_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_78a97d02-af32-35d7-a755-fce1b57dcbd8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ea8333a-da8d-31f9-bc00-a8e453036ce3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea73fcc6-c286-3fa1-91ed-5706c77ad403_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0e05be72-f01c-308f-87c0-9c56a6465876_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_85f315b6-1609-37c3-941e-14ae7936e96c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_33b0aa16-b987-3323-b920-d3665fdaec0f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b03ad8e3-4c69-309f-a960-9926abf605c1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_97fb9bc7-9312-34b2-a2d0-f196c65111a2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_821da87b-6623-3410-b302-a86426e6982c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c6418cba-76e5-3920-8ec1-865f2fcd3754_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_01664ddc-e1be-3241-837e-167a5ea8e5c5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_20c41f4c-3df8-3894-a558-a8202e8354a9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37f05b48-133f-3c14-9b3c-cb31c781aa4a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0392a1bd-a53f-36f7-b64d-959ca6ca99e1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f17fd4e-f481-3b61-8cb8-8ce5e96efe7d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c114dbb5-d8ce-337f-9c70-6449f2331704_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9fd19b8f-97e2-3407-bbdc-f70b9b0e3728_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_43ee991c-22a5-31a7-bb5d-214f86825897_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5da7772c-c4c9-360f-acf4-40de6c28542d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37f05b48-133f-3c14-9b3c-cb31c781aa4a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0392a1bd-a53f-36f7-b64d-959ca6ca99e1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f777913b-b6bc-305e-bd0e-40b3353111e5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f17fd4e-f481-3b61-8cb8-8ce5e96efe7d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_716d8cf1-fb64-3bbb-9778-4c26e7dedd38_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_01664ddc-e1be-3241-837e-167a5ea8e5c5_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_20c41f4c-3df8-3894-a558-a8202e8354a9_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7f0941bc-afc9-349b-96a3-2a6f1fd50ad1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eaafd7f3-f20a-3d21-95f1-caceea180a37_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_716d8cf1-fb64-3bbb-9778-4c26e7dedd38_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc56061c-ec3f-3eda-b749-a904a113a5a0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8c4277d2-8ec9-36ca-9994-cd6e82cb741d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d3b5e48-f777-39ac-b8bc-3b36cfd1d8f4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f777913b-b6bc-305e-bd0e-40b3353111e5_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d754b02-9e18-374b-a17d-149cea3466dc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_072fb7f0-a4a6-3315-8852-f90aae28539c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bee71be5-7186-3557-b0d4-3ae32d9dc9f2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_93d300ff-113b-3bad-b51a-db2e9898b32a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b45c884-2c54-39c8-aa4d-2869f4108b78_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8652fca-7c56-3ca0-9971-c047166d9593_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d77e813-f64c-314b-9586-ad32ec2eb178_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_98a94db9-9c54-32c8-b993-dbfeaa6edb50_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f25fa8df-af58-3bee-8301-aee6b40a0e9e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_554807af-8e85-3634-8f2e-ae12c6ea102b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f1112681-d705-31f1-8478-f0a9a7a675b7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53fd9f31-1a35-3271-8fd5-509c3b0a24be_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_63ba803a-25ac-393e-bdb8-eded325c806a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f27955a1-6efd-3909-a084-94a342d99112_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b59861b9-0190-377e-b870-6398672dcc4e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8652fca-7c56-3ca0-9971-c047166d9593_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1630af1b-8b8c-3bae-82bb-bc6018bd8f95_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c8eabe6-8108-3edc-8490-1c453b22dd44_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f25fa8df-af58-3bee-8301-aee6b40a0e9e_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f1112681-d705-31f1-8478-f0a9a7a675b7_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_138b49c3-b127-3627-b7f8-b655cd40af7d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f3c12cbc-a5ec-30b7-84e8-c9ffcaa84d8b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_825fb325-73bb-36c4-ba5c-2e61d555f5a1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c8eabe6-8108-3edc-8490-1c453b22dd44_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a5eddb12-0669-3bc5-90d6-02691998ca31_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8fa920bf-979f-384a-b8ed-fedbd837ec87_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3fc0ad3c-71ce-3d20-879f-ffea2cae5cb5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_138b49c3-b127-3627-b7f8-b655cd40af7d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d200dc0-adcc-3364-9cbb-69a659214f65_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_261e49ea-2429-358c-bdce-8aed5511a2e2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1630af1b-8b8c-3bae-82bb-bc6018bd8f95_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d59c880-f27a-30c8-a959-032ffdca5827_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_66cb1617-21dc-3aae-a873-aeae7fe59182_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61113ee7-5104-3496-a35b-a3ceaf1869ad_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee02f4aa-a8c6-33d8-b3d2-2a3fa70bafd5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_399e3be5-d40e-3c7b-a605-949ae4d31c11_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_02761161-3939-38ff-977f-e13294081393_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed524190-f906-38fa-8b99-7d2cfa41db66_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_08aff233-6871-3ecd-9552-001d0fba8c0e_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7800dd33-a29f-380f-b21e-44d0460e9f6b_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83685acb-f909-3e08-9964-5dcdc493e075_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e8940e3-7c92-371d-a56b-854da3a52add_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_98c6b1cd-6335-30b6-a3be-c1406ced2be6_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_24ee765a-46b6-3435-b288-e5b49bd92c20_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd0871b1-749b-3117-888e-afd3e1de28a3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4a8a5225-6f30-3a63-ba4a-08cfd15f3352_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_589b2bc0-a519-319e-b849-ff0599b7c61e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_399e3be5-d40e-3c7b-a605-949ae4d31c11_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_02761161-3939-38ff-977f-e13294081393_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed524190-f906-38fa-8b99-7d2cfa41db66_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_08aff233-6871-3ecd-9552-001d0fba8c0e_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7800dd33-a29f-380f-b21e-44d0460e9f6b_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83685acb-f909-3e08-9964-5dcdc493e075_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e8940e3-7c92-371d-a56b-854da3a52add_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_98c6b1cd-6335-30b6-a3be-c1406ced2be6_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_24ee765a-46b6-3435-b288-e5b49bd92c20_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_77da2239-318f-304f-9c32-615267d83c40_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8eee8db6-1435-30b4-b6ad-1acd1a16acc8_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f7f2371f-d595-3767-84bc-7f02a053e664_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_631d7a9e-96e5-3f1c-b86e-c02d17125501_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3f464375-2ac4-370e-a8d0-d29d6c594e82_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7eff3db4-794d-307b-8b47-0b0af5a23d31_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d21f6ff-ddb1-3e15-8588-7a6ab3f133fb_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f0f836c8-f2a0-32ec-8dda-28afebcf8757_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_78d77537-012d-33db-b7de-a417e882a88c_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_59203846-8e0a-3f7d-ac81-5e48ab528b87_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c8347719-8ed8-3a53-97cb-2d2b5ff134e8_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a3611a32-712e-321c-97a9-128d08087088_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_770bfcfe-d4a6-33bd-afac-67c8c54e754c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5c501929-6f8a-379b-a256-47cfe5dceb9b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e280870b-e97f-3b80-92e3-22be001b4828_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c9c3e502-91fc-3827-a549-ac8e13ed8b98_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_820dfba1-2b9f-3058-b4b6-fe20a9bc37e8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2f99835d-f9bb-3160-814d-ece977f7a7eb_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f689314-4ad2-37d6-a687-3e90a4d6185c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2ac354d7-0ccc-3b73-9672-939107c0047b_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_31fafefe-8409-3bb2-a592-ce883b173580_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ec0c9ad-db72-39c5-ae2a-22474c0e42ae_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c28c9f9e-e2bb-3824-9a0b-b50445044122_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d250806b-ba4e-37f5-b3e1-bcf3551e2074_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46e81261-1e9f-3495-b36e-bda2a7bba0b1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f7e10556-1d8b-3f61-82d0-0e1140109d59_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_28207ddd-73b9-3153-b8c5-5b0d363670ff_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_24f820c4-7661-396a-9cf8-759c6e9d2fcb_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d3bdcb6-00d0-38b6-8251-52a3e5cfc1cd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7cb21154-5112-33c3-8265-88f0276f22d7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c41afc08-f80d-3687-9cce-68cfc784a9f1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_307645ed-790d-3e27-85db-174f5baaf937_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_03fed328-8c04-37d3-95a0-c13765e62c34_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3b965c4c-63e3-3816-8e94-3cc9e0b3c8c3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d88a481f-7439-38a9-a503-10fb2655f859_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9af21dd9-956d-3fff-ba8b-389a0a161646_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e27e4b4-b253-3c66-8339-9fd692472ea3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0700e554-dda9-3a9f-aad3-36b7362d1f5a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45a98797-c65b-3bd1-801d-9e2b12b05102_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_98bd9a40-3ec2-3748-ac0c-b6f7a838b357_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3031fbf6-c346-36a0-8ea8-21321d570cdf_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d1c67407-ef67-30c4-98b0-0a72e1061e4c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c6d8d30-b917-3c20-aeb5-c76bfc71e98d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_286472ac-676e-3789-813a-9e01fb303ac8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_427c6270-e7a9-3ae7-b6fb-30ee9cf424e2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ebd1f0f6-15b8-34dc-bc3f-a2072e4fff13_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9af21dd9-956d-3fff-ba8b-389a0a161646_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d1c67407-ef67-30c4-98b0-0a72e1061e4c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83a7d5d9-6d06-343e-be96-967cc58d405f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3031fbf6-c346-36a0-8ea8-21321d570cdf_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e27e4b4-b253-3c66-8339-9fd692472ea3_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45a98797-c65b-3bd1-801d-9e2b12b05102_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0187e46-f307-3c08-86e6-9b10ada21909_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_11af85d9-0747-3071-9e1e-31c3360cf022_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ac3a347-0b41-3c0f-a3fb-ece194b2f899_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_959e91a6-f127-3525-a986-5ba19d93b6e8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ce3944e-c8c9-3e02-aadb-5edb979a1b44_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_848a87ce-5329-3329-a203-1db34ded6c6e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83a7d5d9-6d06-343e-be96-967cc58d405f_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_550c1519-75e9-35e8-8602-cb8974c55efa_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c4d57d2-c45e-31da-8bcf-e4339f0b924b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0187e46-f307-3c08-86e6-9b10ada21909_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d147ba8-2ef2-3b48-92d3-4ced0c65c0f0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de771320-5e43-34b3-bbc0-0f8c15838f6e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_95800650-1cb4-3915-9d9b-afb75f4a7ce5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b4c7a511-ffca-3713-8a14-3a57bebc0411_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5467b058-7c4a-3e57-8b44-b969920512d4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa7f09a8-2565-3976-a204-f342ba1c96ba_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac998d1a-9b4e-3b94-a49a-a20673557bc0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e94c1706-e669-3976-ac88-e346e1cbd456_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_50348808-6f4a-3824-bdd5-332cc50e2385_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45c826ca-e330-3043-8a17-e4e18fd2798a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ebd04a7e-698c-3341-863f-27b8efa286ba_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_624a0544-72b4-3866-a9f2-ede3a8c584f3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de5129a8-609c-3df0-a29b-e5d141eba7c9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1c046a69-03f9-35fd-8aa0-f9819ef47e81_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1e7e871-9e1a-338f-ba0c-a7109391a070_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26709bbe-e728-307a-876d-305c9813c5a7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5b6a4dc8-a00f-3329-924f-68fc4cb53e86_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45c826ca-e330-3043-8a17-e4e18fd2798a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ad23e30-d182-37d5-a1a3-815719b115a2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e4e09bf-eea0-38bd-a077-00ed01ddecac_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e245540-1b2c-39be-bf26-b1f4a702a239_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba414518-4d29-397e-ac10-771fa508c785_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3106eb72-238a-3f07-b093-f82dc6a7174d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e4e09bf-eea0-38bd-a077-00ed01ddecac_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b5b36b19-edc4-373b-9a8a-5525a66046a6_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_425d9ee9-463a-3f79-8ed6-b808db91bbd3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90e25a91-b3d1-3782-86cc-2ef427bb84fc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e245540-1b2c-39be-bf26-b1f4a702a239_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d67d1f2f-7bd0-3c15-ab18-14888b1f5095_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f52f735a-24fd-363c-96c9-7b5249725a35_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ad23e30-d182-37d5-a1a3-815719b115a2_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dfdb70b8-7284-3898-87a2-50e98849db8f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5709037-995a-34b3-a846-15b1d1ac0f69_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4cb3b240-8e24-32fa-9c88-e871cc13c37c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54905c25-7e7c-3655-a8f3-8a9e5fbba883_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e5426881-4c71-3687-86d9-df50f38a4e4b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d8d7576-d718-3937-81a8-b356b8f8a56a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9341fcb8-a413-30cb-9158-d39cff794292_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9c06f02b-0c05-310e-be2b-4f731a205498_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_252ab9b3-bc82-3be6-a5a9-c41e13455dce_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e090c87-33b4-38aa-b143-d0d1f9006615_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6f7ef230-17dc-363e-ae91-884d34cabf8f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5318bd6-eb38-3236-8680-a36ff3849a83_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb2a3fcd-b5db-30ee-b611-e8d80330bd4e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c0e4dce-d7d3-3422-8bc7-3784fded9d1e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d8d7576-d718-3937-81a8-b356b8f8a56a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9341fcb8-a413-30cb-9158-d39cff794292_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_428a0877-65d5-3137-986d-7a54daa3f1e8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8aa6475f-6c83-3588-b7ee-f81b59905c4d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_898af2f6-2bb2-394d-9342-3c1f5ea62d58_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_428a0877-65d5-3137-986d-7a54daa3f1e8_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee514415-4720-3726-ab24-dbb58f15a7c1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cae779a5-9f48-30f5-bc04-743e657d94c9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_35e2b974-b728-37d0-a2b2-0cea4e4ca1f2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8b23020-98b3-362c-8549-deeaeb9c3366_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3e6600c0-c42d-3cac-a049-88b14b9840a0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e57432c1-01e1-33f4-97ec-5c485daf47b1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_69b5f7ab-483a-3a33-b1f8-f70b933a31d5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d45c3495-575d-3898-ab46-3344f59c60cc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de8d3393-b8f6-3676-8a3a-ce2980b5e35d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ef85f19-7e92-39db-933a-5f600a2f522e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6badc183-298f-3c32-9d67-9425bc70794c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_edd464da-7d89-3e80-9553-a59319fa3a01_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e58c9cc-57e1-3bb8-930d-c2e22ef9ae47_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_668be9c9-352a-304c-8c2c-36de679c433a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c84acd8f-96f8-3140-867b-9549766200bf_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5a5d0056-3adb-3b18-8826-40c5beb6624f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_70c4b034-c64d-380e-82be-d60b05147919_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b4a8321-bb5e-399f-b98f-7f9ec4968705_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a7007052-d221-3d81-9cc8-fcbadf23fc7e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eea122eb-636c-3d2c-a9d9-a652541da5fd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_05745501-b2e7-396a-8ec9-992988117082_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bf4e470c-ff65-30ed-a90b-41502bc4ca85_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_35a00b90-ae8c-3b67-8373-ad644b9e119e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b4a8321-bb5e-399f-b98f-7f9ec4968705_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d18ac955-cacd-37f8-9df1-b763c0e22f1c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8c062ceb-4ade-3642-8b32-2e0b8b802714_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_307be909-4aa8-3f13-9a14-6b024ce72579_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e350a7d4-b124-3768-a182-61354943bfe8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1e58ab80-1b1e-3074-9f45-20c37d1d24fe_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4aa1a01d-50e5-39b9-bc42-751d3171f2b3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9341fcb8-a413-30cb-9158-d39cff794292_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d8d7576-d718-3937-81a8-b356b8f8a56a_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ad23e30-d182-37d5-a1a3-815719b115a2_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_428a0877-65d5-3137-986d-7a54daa3f1e8_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0187e46-f307-3c08-86e6-9b10ada21909_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45c826ca-e330-3043-8a17-e4e18fd2798a_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e27e4b4-b253-3c66-8339-9fd692472ea3_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45a98797-c65b-3bd1-801d-9e2b12b05102_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3031fbf6-c346-36a0-8ea8-21321d570cdf_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b79fa3f4-90fd-32b7-bf88-0a58830d0446_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d96efbe6-68b3-35ff-a070-b5814bba22f4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_337335c0-e329-3363-a596-0d601287e21e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9341fcb8-a413-30cb-9158-d39cff794292_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d8d7576-d718-3937-81a8-b356b8f8a56a_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ad23e30-d182-37d5-a1a3-815719b115a2_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_428a0877-65d5-3137-986d-7a54daa3f1e8_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0187e46-f307-3c08-86e6-9b10ada21909_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45c826ca-e330-3043-8a17-e4e18fd2798a_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e27e4b4-b253-3c66-8339-9fd692472ea3_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45a98797-c65b-3bd1-801d-9e2b12b05102_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3031fbf6-c346-36a0-8ea8-21321d570cdf_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1e440a43-ba11-3dcd-b52a-f8acc26838aa_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3646dcd-3241-3c8c-878d-ab2402b26242_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_50d8cb51-8b71-3451-a732-90b29022a903_13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6cebe15-26d1-313c-b185-1eae6f58f1f1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b138ba85-f9ac-377b-8cb6-78f4f4aa92d0_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54a4f43e-4437-385a-8a28-62a6c718c800_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4342f023-6cf1-3350-9c99-e063d4883409_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82a14373-39e2-3144-83fb-567258571ddc_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_88550324-121e-3366-869c-015e166b8d0f_13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e03ec9ee-bd38-3d6b-b407-90277719e7a6_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_18c7d81b-ea69-3e28-87e1-007d3198d9fc_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ea9ed13-0a0a-3b6a-9969-1babc73f24c0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1f41d239-c5f6-3324-bb09-492b0662bb65_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56191e2b-d469-38f8-a19f-201cf74d69f4_13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_516599d1-dd8d-3fda-8aac-d5fb968ae4da_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b77f1e16-949d-3da6-b142-256c239f6e8e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61a33fca-3fac-31c9-88ca-03a20a26323b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ddfd98b-a236-32c9-bf45-131dee22096e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_849a8627-b931-3dde-b913-78af56ec6690_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_84775469-b0c1-3398-845e-1b2c152956a6_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0737dd52-fb1b-3ffc-b1fb-60dcdd702cae_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_514a4081-a0b1-374b-8505-fa2eea9810e0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00a60446-aac3-3f9b-b7e2-ecb7222841a5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54c26e5e-26c0-3e59-9ffb-870ce434423b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_187408d0-9b1b-3c1a-89b7-6a454a419080_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c341d828-5929-39da-93be-046eef2d7ee9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e63c6818-27b5-3cf2-a8bf-289bae692023_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a3b0542-9399-369d-b929-f2130e568ef5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7078f00a-0378-33d6-9d54-02688fbdba04_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bf4c73f-a895-31fc-80d8-759a4e951c44_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a3d7bd46-5534-306b-9273-1e09e678809b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_92038338-5677-3d06-86eb-88e957445524_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_20612179-be65-353f-ad63-279f05e7fb1c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12527e92-cef9-3367-8525-b736a1a6883c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37e56553-9368-3c40-952e-2511d7774c7c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00a60446-aac3-3f9b-b7e2-ecb7222841a5_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_86156a47-4cb2-36e0-85e6-6e62cf42b933_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a3b0542-9399-369d-b929-f2130e568ef5_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_187408d0-9b1b-3c1a-89b7-6a454a419080_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1d1796e-1f11-3144-86df-f573db2b5eca_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e0d8df1c-a6f8-3193-ae3f-5f05ffa381b5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b2d644f-ed58-350b-9daf-ed43ccc21606_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d65c13f-25d6-372a-835c-e47390b020c0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3257c3ed-eef5-3a70-917d-1ae71135456b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_86156a47-4cb2-36e0-85e6-6e62cf42b933_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_35ac8716-2ca4-3fff-a494-2e2bb29aea8a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7894811a-4701-3ac0-9c0b-a3b6bcdf3877_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_03545356-95e8-3bbe-9378-ca3c987819f5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_28ad19e4-502c-347c-8a78-7efc3e49a526_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c577533e-47d5-3d4a-9bfd-b3847508d4d5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e06db697-38fe-3f4a-a98b-13b2e3555f46_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab6f8f4c-ed9e-3159-88f2-72123f6a5e14_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91ec58f1-f76f-3421-91e8-c77b948525bb_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0adb4485-d30f-3320-86fa-d8c11755b3bd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9c6ce1eb-7051-3bbd-ba85-0e7dc5eb2db9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_21261689-a869-38b1-bad2-5e2997be512c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bcfd4a71-de0d-3bdc-9baa-696e3110e41b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_48619964-91ea-3121-9a61-9a7861cda3ec_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52f5a19e-af90-3eab-91b2-d28b5ed60110_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee2b67fe-fc26-317c-bc78-e5955d7813aa_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_201f6e56-d9e4-3494-8dca-ac9e68cc465d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_465ad7b5-031f-3f71-aa50-a426d130c0ee_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4cbff44-239d-3589-9971-f784273cc576_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_85012b8d-dab7-3684-9e80-198fd0e4c033_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_99510d06-6eeb-30f7-8794-ea7e364ed7e2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_465ad7b5-031f-3f71-aa50-a426d130c0ee_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c11f845c-4284-3d1c-acab-faad6833ed13_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_95a19580-7ce8-3660-833d-73ec12f4b336_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4713f197-eaf8-3275-ab25-322b9cbb5730_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4cbff44-239d-3589-9971-f784273cc576_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a432210-9194-3a09-96a8-08e1d4f82733_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_03f35725-4050-323b-b6a0-1b143300df9e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2202282b-3a62-319f-8b44-6b530c1dc84b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56f66051-e349-396b-aee4-e48fe1cd3c46_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91844293-e499-3d03-9ecd-45466e7f68d9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_71172c0e-0166-3ab0-90d0-7ae741237609_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9547978-60ec-33e6-8467-49a713cce878_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_285f4781-59aa-35f7-9541-37b4d5433853_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2b412e6-9476-3849-b663-a8acad10747a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d251665-b0ba-38e5-9dd1-c2018a333388_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ce2b998-0001-3f4c-86fc-ed06f76915ee_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b99d5cda-178a-31e7-8148-73583d114bca_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cfe42e52-2fe0-3f35-acea-9721b09f4246_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b2f5ab12-d932-3f4f-9c05-731abb5c4fa3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d700e5b6-e5ca-312a-ac44-b96546c8d64a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3afdd34b-7e1d-3564-8c7c-f81c39e60f1a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de8b1c3f-8d25-3fa1-ac50-35c3ca14b825_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6dc89e1d-b9dc-35a3-830d-e5c4a4fd5c58_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8f570d6d-0edc-3a94-ab7d-f82e4065927c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ca10156-76f2-302b-8713-5bca4ee4d823_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_803ae06c-aec9-30d7-8fdf-e4d747d1a9c2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c37b5616-1d7b-3ce0-a1af-8d1071d53328_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5bb59c6-ba31-38ad-b244-6ec1dc09b28b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6eb00f9-b982-35d3-a541-421ed231cf3e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c3dd9f64-fdef-3510-86ca-0d47305bc781_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d20dea6-9b96-39ad-91db-a4e84f5e0d93_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1a54e228-b0a6-3d26-85ae-8c8f87efbd09_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_92ec1910-dfc5-3385-be01-4e97b85e9de2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae9f998e-2800-35df-b8dd-925758685b93_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a908c3e-9dce-36f1-afb0-3c7653cd1310_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61fef315-3c8f-3299-87bf-50fb55e10952_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aa0c058a-113f-3fa8-a7a9-8b294a14878a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffa2669d-4285-3366-aa5e-6957e13c0670_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b73abf65-2e86-32d4-91fb-bbb59544618c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d63a981-d10a-3781-bd30-a09c8e5ccf14_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b0d16b5-0965-3f56-9002-7685f4b08aae_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7a682f05-19ed-3b17-91ef-fd9ed73f6d19_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49ed8429-bdd1-3f60-820b-5a2614e9e235_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c45eecd-8ce8-3457-98b6-ebeb9ef07456_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_40739d98-ee9d-3532-881b-dc3938c2442e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bdec17ed-d784-37f8-a38d-7ff5bc184bf6_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f8f8d036-73d5-3080-a00d-a6b375dd5fef_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1d160f-33d3-3411-ac18-92700924fd49_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b174420a-db69-3b25-97f7-ca201dcf1671_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0773961b-c356-31f9-825c-54217c0fc0b9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9a4dd14-f4be-3130-a14d-b7beca0ae96b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_06fa6a93-638d-3ad6-af51-b750d836325b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49ed8429-bdd1-3f60-820b-5a2614e9e235_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc1c4547-8b56-3450-bb48-4c5bc583278a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1d160f-33d3-3411-ac18-92700924fd49_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b0d16b5-0965-3f56-9002-7685f4b08aae_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b630566f-bd03-34c2-ac03-09899f86dae4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_58333c03-948f-3120-8b63-a0cae0dcc777_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_24bf5af8-2f8a-323b-9848-ef9b42050386_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc1c4547-8b56-3450-bb48-4c5bc583278a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12757c0b-1bee-382a-9370-69075a3765bd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dda51557-ffc7-3d06-9580-9d7b1e3dd3a7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea6f11b5-52bf-3daf-8c4f-9aeac9618b8c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8262970-6a4c-3ba0-b7da-74b59e90df15_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_125007be-797d-3b3b-b76f-5238e6805940_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b630566f-bd03-34c2-ac03-09899f86dae4_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2603b589-0235-3c10-bce4-0ba6bffc0a53_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9de209dc-d788-3f84-94c3-62327e782c14_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5b0fd2b8-01ac-3063-948b-6a288ba598a4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5e2abe11-124c-358a-8644-ec829535d5a0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54cb2ce8-32d6-3e01-8c1a-f19d79453402_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6f77334e-6e3e-3b3b-81bc-c12ddccf6c8e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d311e82e-a3c9-31eb-9329-047e62566afa_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ca1d156-d8c9-30ca-a0f0-020350708338_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a6167359-38e2-3abb-94e1-cf2ce04f6a85_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_138c3761-7e2b-3aa0-a85d-4a8c5461234d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f237fcb5-5834-31b4-872e-7872f7016eab_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b8596c0b-82e2-308f-9cd3-3003194cf030_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cf217b64-3ea4-3f65-977d-47a7a0108eea_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e42ec2ae-31b3-3488-9a00-05811200c0c7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b101853f-73bf-36f5-95b0-3fe25fab0677_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23491052-0e9d-3337-899d-b7993d7f6638_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1d160f-33d3-3411-ac18-92700924fd49_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b0d16b5-0965-3f56-9002-7685f4b08aae_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b630566f-bd03-34c2-ac03-09899f86dae4_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc1c4547-8b56-3450-bb48-4c5bc583278a_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_918a96a8-500a-31cc-ad2b-5bb6aa44da80_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3c8eab0-7d49-3039-b447-0a58b1cb6b85_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54d6e9f2-0777-37e5-b350-817f61ddc312_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_93d19720-b9a2-3338-9800-08e2eafde93c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f0c83e71-9d23-3403-b35e-e297ebbabe97_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d4414abb-ac99-3c1c-88d3-30efdf504f65_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab5a3d9f-ccb3-3e9b-97d9-b4af5a8702db_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34721174-4abf-3077-86f2-2e486ab02a40_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5271089a-10e4-3eb1-811b-f9e0abf8dd0a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc43a166-4199-3940-a041-36c47a1fbcb9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1f3f40fe-711a-3bff-be18-cb93ddb970f7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fd3b9cdb-bfa1-378f-8abb-3dbd92155e70_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5815dfce-c471-3d4f-864a-0ff7b333bc8e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2cc8f722-fe68-3bf3-a3b8-5b8861bd2fec_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d4b4e967-bd5c-38f0-913c-60d2020e804a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_187408d0-9b1b-3c1a-89b7-6a454a419080_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4cbff44-239d-3589-9971-f784273cc576_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1d160f-33d3-3411-ac18-92700924fd49_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b0d16b5-0965-3f56-9002-7685f4b08aae_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b630566f-bd03-34c2-ac03-09899f86dae4_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc1c4547-8b56-3450-bb48-4c5bc583278a_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a3b0542-9399-369d-b929-f2130e568ef5_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_465ad7b5-031f-3f71-aa50-a426d130c0ee_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_466d8699-b90c-3be8-adda-e1df0c208256_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8fe43f3a-dc28-36bc-b27b-4cd39034a39e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6bc54c7d-f0e1-3d84-a7bb-2fde6643e07f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_187408d0-9b1b-3c1a-89b7-6a454a419080_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4cbff44-239d-3589-9971-f784273cc576_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1d160f-33d3-3411-ac18-92700924fd49_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80029533-0e8c-3888-a516-697ca1e46948_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b0d16b5-0965-3f56-9002-7685f4b08aae_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a31fb0-910d-34dc-bb86-4a8038f11737_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b630566f-bd03-34c2-ac03-09899f86dae4_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc1c4547-8b56-3450-bb48-4c5bc583278a_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a3b0542-9399-369d-b929-f2130e568ef5_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d71e079-cd50-3e15-ab90-3b634a588c2e_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_465ad7b5-031f-3f71-aa50-a426d130c0ee_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_13588034-af53-3681-86b0-4e70d93e058c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_50d8cb51-8b71-3451-a732-90b29022a903_15_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_15_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7281a39b-bdfc-35c7-9df8-f931d754d099_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_88550324-121e-3366-869c-015e166b8d0f_15_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56191e2b-d469-38f8-a19f-201cf74d69f4_15_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a3b0542-9399-369d-b929-f2130e568ef5_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f0861df0-f052-3d22-9be7-e5066e7bc6ba_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a4222c0-3470-3f85-be3a-89ba8d19fc93_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_576f35e1-9b3d-3d2f-838c-0643279498f7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_498dd2e9-7b5b-3fec-bce6-e8fa2fe85e9b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_36be5676-541a-3522-aea4-84faab5806d7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8c72ca46-04f2-3279-afb0-678c829eb288_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e257015a-99f7-34df-8333-d2cecbd52ba5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e724c387-f73e-31af-8200-f84727189def_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_667e1cb7-c64c-3e3c-87c0-de33df801e67_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_10de7e59-f8a9-33fd-b17c-e19c3f0bcd47_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ff4d069-4654-3b26-80e7-da2dcbdd664f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_64475f0b-1ee4-370d-827c-842dba53904a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8050b2de-f03c-3d81-a603-f7e5654c1956_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34f49f35-e8aa-3633-abdd-46f882384727_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34e4cdec-84d3-35dc-9191-3276f0d47f78_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e29c6743-c44d-3168-be9f-6f9138decbff_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5262e57e-6de5-3bb4-ae48-1c09d9ebc282_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_677750eb-9ba2-3d63-a174-5c21f65a091b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_93b1a403-40b8-36b9-a005-027b8b4a7015_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5040b26a-bf36-3d82-8e2a-dd6bb36ec5d1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2f61921e-4112-3467-977c-2f51ac58f23c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34e4cdec-84d3-35dc-9191-3276f0d47f78_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb453d2-e336-332e-af0e-8690b0771a9c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5df9f690-ae30-345e-843b-dd81e91b12f0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e297fc0-1f2b-3144-a474-11183f5b3e0e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5708656b-f7ee-3846-b94d-5d42eef64047_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e7c95f2-a85d-39b9-a68a-4ed664393e0c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a070b04e-21af-36f6-bd8c-1479575f5294_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb46db5c-3ad7-3eed-8e76-fc76b534f4d7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5df9f690-ae30-345e-843b-dd81e91b12f0_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c3915473-c347-31e1-a2d2-d2e6543075c8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb89291-764b-3aca-8390-718f11c15fde_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb453d2-e336-332e-af0e-8690b0771a9c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7767e499-cd03-3480-923f-0cfe46e0e9e4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_acc547af-f298-3885-8682-d4a8201b12a6_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_57b0bf8b-3fbd-3132-82ba-e064660d1180_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_254a663c-4562-3a3a-b50f-ffc1193c0a83_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f3287bc-8f12-3e4f-9481-e0e5c4eb7b19_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c74b820d-e740-3f0e-a9e7-9e5fa7218c17_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e30f261-e8ef-353b-b5b2-f827b17761ba_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cb64de54-5586-3030-9d48-5115311b086e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_50046e89-cce3-3198-96fe-05073daa01d0_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b8ae1af-b6e3-3ef2-8b86-aa0fdc3e27f3_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aa9f91b0-8df6-363c-a978-a11660a5c7ff_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ff3ae573-f55f-3af6-80b9-732ccc6f7aa1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54d42c89-4627-3596-b06e-6ef2cc63b06a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_09923e5f-e976-3740-90a9-b8d85b66d21c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c74b820d-e740-3f0e-a9e7-9e5fa7218c17_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e30f261-e8ef-353b-b5b2-f827b17761ba_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b329c2cb-33e8-3a9d-9340-f26cbf8586b2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6bdace42-2db5-3071-9f7c-e2ae6882fdc5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7b6032d-ecce-3ca3-b5c8-2caaa454f899_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc745182-7a6b-3403-8ecb-bb0f9c15d0a8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa9c24da-8d77-3ff0-bd10-94db58a745f7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d358f2c4-c860-37e6-abf2-ed5486c43bdc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c42a7195-a838-3939-b1ba-ef4a18ff6bdd_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ffe6802-1700-31b2-8a61-dd10b000b4b2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f3eb771a-2ca8-3ea5-b07f-89a7ac4c56f2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a6b37d65-76f4-3aa3-9e66-89508984034f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9728f385-c236-3261-a0f5-7f9c87909a3a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a5fd9606-976b-357b-8018-f2d6d6002f17_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56e92269-d480-3d6c-8062-598884ec7a42_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_97f88eb5-156c-3868-a4c8-45d44735bf33_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ec12d1ea-c2b1-334c-bc37-968d3e0b2bf8_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b7c5b797-cfe0-3bef-aee3-ed44647c1163_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee079adc-bddc-3409-88c6-b74f40503ebc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f3d93d81-9b6e-3f82-9a98-76c03a06d4c7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e43d15e0-22b4-30c0-89cf-2dbc7823577c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26221b16-0e9c-3389-8e58-89472e334043_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_29da0d83-c2ec-36b6-8a7e-1b8048ca47a2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b869bbe1-5d00-3234-9a18-2c1df1515699_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_73df9dbb-510f-37b9-b5d4-f2f1ea3e89c4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3c3ca39b-9711-3c78-8f2a-b4c94a625cf9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_332d792a-a87a-3060-8919-d58e4a93d3a1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82d50c47-8ddc-3cee-a707-89c64d7ed6b4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26221b16-0e9c-3389-8e58-89472e334043_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c10171a-7bcb-31e9-8448-d96151e14fbc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fce469e3-e0dd-3d9a-872a-5b823ca67524_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e43d15e0-22b4-30c0-89cf-2dbc7823577c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b869bbe1-5d00-3234-9a18-2c1df1515699_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ff8eb85b-8637-3b61-97be-222ad769ec46_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23ae6868-110c-38f7-b9b3-bda3202ca5e1_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c10171a-7bcb-31e9-8448-d96151e14fbc_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a89f69e-90fb-3444-b8fe-4d618f9843c2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e372ef18-8c80-3d55-8724-84b012a3147b_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e7dc2624-40f9-344d-92e9-7be3c412a301_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fce469e3-e0dd-3d9a-872a-5b823ca67524_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0cce9f20-914f-3fe9-ac1e-fb603b3ad2a4_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a99f4a35-3daa-362e-a8ec-221272f4596a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba685ee7-f3fa-32c9-875c-545da162a443_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8cb46a61-973f-3f6c-b221-2c8923bb8703_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0deabf46-f1c2-366f-8462-5200c1dc9f02_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b8da3ab4-1e3d-3b44-8f71-f039be064db7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5120c48d-6b45-3328-b3c6-79781f581f2a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed2d4317-72b9-3395-9f0e-782d40b0657c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc2196d7-2c73-3666-a264-ef5852c88fda_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d542e9a7-a4fe-3989-a7c2-2250d3c3593e_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_51fb278f-4bc9-3a24-933e-e925d1b65788_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb06791e-0652-390f-9cae-d66726b8898f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9b7eb2d5-e465-3bc2-bbf3-d913fe8f3b95_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea208300-9562-3d7d-a5c5-ef7699f47021_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc2196d7-2c73-3666-a264-ef5852c88fda_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a146582-ea28-39d3-a098-b9884285bf4d_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_73e0cb82-c093-3597-85d6-e3539938cd74_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5120c48d-6b45-3328-b3c6-79781f581f2a_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4eb1b9e7-a1a8-3b22-8714-df204b37f6d5_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_081aa0ae-d050-3d6e-90ac-b896c4c18f6c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a146582-ea28-39d3-a098-b9884285bf4d_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d53e0167-62a9-3b2f-93c4-deccc1ff8011_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0aaf1fb3-0ad6-34d0-8c97-57709fb66f8a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00ef55aa-ed0b-39e5-9230-e5368f51c348_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_73e0cb82-c093-3597-85d6-e3539938cd74_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_99fd57a2-aa0e-3d38-9486-372c7aa85413_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_96b4c3f2-1fc0-3670-a108-666db37aeea9_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_22cae7e2-9878-3a17-a4b8-d6ad4cfc9ebc_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91a57a33-e368-34b3-b324-730fb1268d75_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8f2b205-1abb-3f53-ba22-5b5c26e77440_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_78d2c8a3-5363-3578-8bb2-5c18ca2c308a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_66c56d1d-bec9-326b-9c35-3b71f47feea7_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c120ace-834f-3828-88ea-e64dc5910348_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fce469e3-e0dd-3d9a-872a-5b823ca67524_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5df9f690-ae30-345e-843b-dd81e91b12f0_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34e4cdec-84d3-35dc-9191-3276f0d47f78_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a146582-ea28-39d3-a098-b9884285bf4d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26221b16-0e9c-3389-8e58-89472e334043_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5120c48d-6b45-3328-b3c6-79781f581f2a_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_73e0cb82-c093-3597-85d6-e3539938cd74_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c10171a-7bcb-31e9-8448-d96151e14fbc_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee079adc-bddc-3409-88c6-b74f40503ebc_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb453d2-e336-332e-af0e-8690b0771a9c_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc2196d7-2c73-3666-a264-ef5852c88fda_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_55c2d0c3-2d20-3a54-ac21-f92b0296362f_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61db4a67-a8c5-3629-81fb-cef4d4dfe944_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aab98cb4-d481-3f43-ad1b-0e43adb7b92a_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fce469e3-e0dd-3d9a-872a-5b823ca67524_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5df9f690-ae30-345e-843b-dd81e91b12f0_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34e4cdec-84d3-35dc-9191-3276f0d47f78_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a146582-ea28-39d3-a098-b9884285bf4d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26221b16-0e9c-3389-8e58-89472e334043_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5120c48d-6b45-3328-b3c6-79781f581f2a_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_73e0cb82-c093-3597-85d6-e3539938cd74_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c10171a-7bcb-31e9-8448-d96151e14fbc_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee079adc-bddc-3409-88c6-b74f40503ebc_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb453d2-e336-332e-af0e-8690b0771a9c_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc2196d7-2c73-3666-a264-ef5852c88fda_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_74ced5b1-cbb4-3f68-9555-8fc569fd6dbf_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3cebb160-fb56-3c6e-babb-13d9ba2d1d43_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d137191a-df10-3706-ac81-017bfc89e829_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d16e74a-753c-36ce-89b7-033a935a8489_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_31a10454-e1b2-390d-bf4a-327cea6a3bee_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fe87f7d3-7d04-3c91-a1e9-67e3310c2229_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16d89b40-06fa-357c-bde9-4c060b2e881c_1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a077b8ff-1b70-3afa-b857-f6ccd2b77102_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_22f17c59-e1ad-3366-8a19-25e286e242c3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d3fc9e3-a1f9-3df9-8022-4d6a17c120f8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab04a4a9-347e-3d96-81c5-fa22fc6072cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_770cc26c-a1f4-3871-b282-88b258b6258c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c9718dc-da95-391f-b01e-8f38d4789099_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d6c3c01c-27e5-3001-a37d-cd478c79c65e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b5d2312-142d-3a4e-b6f9-2b3818903de7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_922a56dc-6b19-3b91-a679-3a1fd898b805_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_241b6aaa-8d28-3d6e-9a63-38a4331a3e89_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fdf30f45-e230-34f1-8e45-fcaaf0d5011a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ec492d28-cd06-37dd-9f21-44a31d356f5d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c00ef574-a1ee-3083-b39b-c40921ed9527_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6701b029-dfe5-398d-9b62-de47b362c9ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1cdb2cd0-745b-34e1-b950-0e7aaa31bcdb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_927a5101-a802-3ffe-8b9c-0ccd78f9677e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f39633de-2a89-324f-bbf8-6304f3c0e516_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3fed372-2bf1-3eb7-8f11-521aff5d5bf1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a3c85b11-e70b-3a02-a798-252467b7b395_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90565092-7a72-3e5e-8aea-b9745f77b033_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a5beb6a1-dfe7-36a4-9e7c-d6f2276190c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07c60fe4-e13f-3cd2-a567-b7d2323cc1c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cf2e1c76-82a1-3b2b-aa8d-71a7358d1458_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d23f3916-e032-3b6b-a25c-e19786278268_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1b33fc29-546b-3ee1-92b3-64a24b954b17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91716a8c-93a2-3259-a786-824e4923e707_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_efa69277-4fdf-3bc6-9c47-de49b37f6df3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dcda6558-cd78-358e-92e5-31a144350332_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_96ffa2ec-6031-3988-9f7b-43505dbc1c68_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_df1e5feb-c57e-30ab-bdc9-0a2d5ef70b5d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_868ea417-77bd-3b20-9a6e-1978964f4b77_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_712b0357-7bee-3595-b59e-c37f48bdb877_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23e0a7a3-0c8f-32db-98b1-4faead3e14f5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea6c8612-dec2-36e0-acb2-742498e5cdb1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6af577cf-192f-3e87-849f-373e01e030e1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f920300-257d-31eb-83e9-1bc434423e86_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d56a95d4-adda-3241-b483-db5601241f27_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46a54e0c-9fab-3d17-82bd-751e4e440ed8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5f0bc24f-0bc5-3615-b008-ad94f8557e8e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_671f29a4-4e1c-3482-9689-3fed2313f452_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38f94e52-8ced-30c6-b4f5-48b2799a62fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c489c52f-56b5-3412-a21c-5d67aff4eae4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_daa9c25e-b8d5-3307-a490-b8e414b16155_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f582a2fd-64d8-33dd-8662-e4223ac90c1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef560b84-a939-3a38-8c9d-54048ab02721_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9c866c6-10bf-3bff-8291-258c2a6f0ca1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8842d7ad-d5c7-3df7-bb73-b410a8bf0e0a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d92504b-c338-39d8-958f-ab6abc12e4a0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7c6a15e9-935c-3ccb-928a-56dd18198c71_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8a4caec-df5e-3d29-be6f-59e6c0b9f3f2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f6fb28d8-9384-3d70-8a5c-56f260c6f571_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a19f38bf-4a0a-3427-a6f2-f2db63de3276_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_802db03a-507a-33d0-bf80-cf5ca8d49fcb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0d864beb-305f-3bff-a920-04c177683c55_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4a98bb81-d193-3a8a-9509-4864d7cc472b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_602e4d8e-240e-3f29-b19c-9480ee591a8c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4bf09c8c-cce3-3585-901c-f2afaaf70fa5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fd598991-e252-399b-b157-0ce36664c235_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_20d928b2-6a48-3109-a364-b2102397e799_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7ebd1cf-4950-34d6-a3cf-dd6f023e2f61_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53501224-a229-39a5-b1fd-33442dbc8343_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa4aec29-dc6f-3071-bb7a-f28e18532e1f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc3133de-c1ae-3231-9a95-de52a811fbbd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_716ef1f8-de04-301a-b586-14fa7efd984a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2645bc3-3d7e-3ecb-9954-1adb1976cc7e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3e2b1b8-609e-30fc-a903-aaa59a94fca9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d5cccca-51ca-3646-993a-65fea52c88c5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7c3597a3-9fa4-3c1e-9c60-5bf51fe0dfb0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b0bd427-a612-3423-a20c-04a10381eeee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cdba5d5e-a557-3987-a61b-9d5256bc66d7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_36156937-9718-3f67-a8c9-0c7335ada278_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa31e541-9468-3413-89a0-9e9c48155b56_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc43e129-0b7d-327f-8ebc-0ed1bb0e26e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_830b0eaf-3b86-3276-bcb9-6699e2eeb124_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_48d6ad6e-1dd3-3486-b25e-df12ca5893a4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_344df115-7168-3219-89b2-4a15ef4e1a6a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed1eaf6d-c755-3f84-83a9-044fec9985f3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_499d0bdf-70a5-3d9e-83f1-efe2646cdb3a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_360a2340-e335-3e26-afa3-01cb65827bfa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da4339b8-9f84-38cd-84f1-c4d725ea912e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2c755c9d-ff47-3a47-81c3-b57857830945_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_47308f5d-0eaa-3833-970c-8b516a8ebf4b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b96387f2-3230-3429-8d99-eb264bbaf2a4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2f432390-dac6-3fe4-b7ff-01effe388b0a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f62ed523-6deb-3f0a-a119-faf23d263e11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_86b42feb-fc07-3df5-a6a0-1a5954bff810_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba19957a-cd21-37e4-8b0c-ee5d35e54fc0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4f39ffb9-0617-3713-8e56-e50cc636eb9f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_345352ab-6db6-308c-b1f1-ffa3be3d73bd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d96c70ff-4a31-3e3e-9e93-44fe3eab4f40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34fb259f-4982-31cf-bcf3-b28e1c33febb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3035fee8-5f04-3055-9148-688e0c57e85f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_11fb88e8-71af-37bf-8019-4a5a3faac29d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_68220337-1552-3c5f-9546-83bc24af7dab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_471f1d9f-0441-3593-836e-509787633db1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ec1b7a55-5e1e-3a2e-84c9-76b52b44567a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56e06932-2aed-317f-ac28-a511ccaecf6c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_51b972c5-ef90-327d-88d2-c259408ea78a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_33eb9f1c-6620-3a1b-977a-1b58c8e80437_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_603f2506-2a21-37e1-9395-dfa0d5c10160_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3e8c21b1-5a1c-3feb-bb0c-74b7c2dd954a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c1bbaf0-48a6-3b83-a31b-3f8014a5ebec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_150a2b64-8325-3180-ae33-c8c1c413a6ca_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f530e6a1-1d20-384f-ae60-068987118edf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cd6852b1-0afb-3d58-927d-148cc16ba253_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b1ad2ab-f4ee-3bd2-8f07-3b9bce20415d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_47ad8f8c-4797-3dd4-9e42-de7877c0f129_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8332c60e-1f0c-39a8-a906-c3e2f66c8d23_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8b109257-f015-3154-8d38-b903dc9e7389_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_993400e9-782d-3d6a-8529-754414e4fdcf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a3da4108-6bc4-33a2-8a72-5fc9611be43c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8833e44e-358e-3a36-96f8-01b3596a93c2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_068975c4-867e-35a2-99a6-39df9c59ed16_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a35ec56c-ce20-36c3-9d48-ffdcf28d0c70_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a2e9a223-366d-3867-9a58-05ae3c5b0362_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eea5884b-2dbb-3dd4-b69d-2d662730faa2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba025343-5694-3eda-9ee9-48625781bc9a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fccc0440-567d-3869-a93d-98680934d06d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_36861b9f-4396-3563-9e14-618d4cfbff55_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_429180c3-337c-38aa-8abc-74cc7af82c00_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eeabf8a0-0d09-316c-ada6-77f368b6e90f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a029eab-9848-3006-b219-773dda8d3e1b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_02b09316-82a6-3e25-8797-237ec7bdda9d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fe79a959-9c74-33ff-aac2-340928b451fc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f1447238-4bcf-30c5-9287-e9b746450378_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_87b4365a-02d2-3b93-8d72-3c17254d05ba_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_541f3907-8e40-3066-9d39-c1e19e421f5b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9dde58f8-5dc3-39a8-99e2-fec843035e02_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e85a121c-c263-3210-b559-889226700b9b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0da1da45-fb3b-37fb-af28-cb57f726951c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd8e8fe4-c0e3-3423-bdb2-0745e1ac907b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3e0211cf-f4c0-38e0-92ac-1cd2b3343a78_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8ed68d7-171f-38c1-bd00-76814cfb7560_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de76088d-7712-3ed7-aaa9-6e6136a46c4c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9bdcaf66-9c8d-3adf-81f2-ee0679663f4f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7737b1cf-080e-3a7c-b9fd-e06f973489a5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5daa1a9a-2443-3053-9263-fa2b28dccffa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1bed0fe5-d927-3c60-8dee-ef69187f1e5d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a41ebb96-7d0e-3e9b-90c5-405a0881bc84_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1bd09113-5554-35e3-b87b-6f256b49fcaa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1e593297-8fab-3945-b46b-6846d2798ab0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1cadbc47-f831-37f0-855e-93fa45c23fe3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd428243-387f-3f32-b45f-b585a9441a89_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ee804da-7195-3bf6-ab42-e8cb5970ff49_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7f2a3373-d969-359e-b2be-da9de945bbb5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00455151-b5de-3dd2-84cf-514b31608ca3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5c8f1ba1-f895-34fa-ac82-cb66437bb304_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9a2d046-5dd7-3543-868e-1acdb9cd32c0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c0c2baf-eb57-32eb-b03e-212d640a8e9f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb720593-1af6-380c-a0cf-4d7e7630f945_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eaecfc8f-888b-39be-a039-05a9d919e2b0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6483756b-a335-3ccb-899e-21b8fde001c9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7569b35f-ec21-3b2e-93df-07042efe27e2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_791e54c0-c958-37a9-8807-b1b76c209e05_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c3bfd019-ce7d-3864-a482-e8752e32f2f8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_22ddfe20-46c2-3362-95fd-92df06a3c76e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6b9f3fc4-0d00-354f-8710-0d93b67fcc92_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fdb728a4-43af-3434-9d39-a798cbe86e70_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7a1ac939-5564-3b8d-9e7e-b0092529a1df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2175f32e-288c-3f2b-b934-67562c35c1ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f119f2ae-2aca-3060-a994-d3087584f662_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0826d052-5a07-38e4-b1ce-329fe8545a06_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e4cc091f-c06d-3751-b45e-b2ad3d628117_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4920733f-70dc-3ea0-9ed5-d2b6045b881c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8f1df067-31ba-3da1-be15-3cd11047c4b8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_44a3f05d-fb08-31b6-904a-be273e1d65fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a3eab5fb-087e-3123-93dd-26e1c455eda0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_635231bb-d7c7-3eeb-a98d-a36a2238e2fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9c0c072b-0bdb-3a2a-83b3-d4b33dd77990_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7ab2408-b8d9-3deb-9c9b-bf8a1ea20d9e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae15afb8-1b2d-3aae-8200-4b5da537b53f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8ac3b40-b1f2-3f60-9f9f-a5b07e7a7a53_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_33fd44c7-53af-32ef-8bfe-012d2d21120f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83059cd4-615a-36b9-9ce1-734f16b07d87_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7cfdbe93-c35e-3d58-8dbc-f9653be60a34_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab5ced21-1d10-365f-85f4-86795db93502_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd27d8bb-7adf-3f91-ae99-621e476f5523_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5afb1408-f89c-3b66-918b-422262ffb72c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c2c42752-237a-3604-9401-9b8e2b69a93b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4f8ef84-b7e9-3612-b51a-f87addf9d65b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_72135da7-d410-3aba-a825-64b352199f98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a1ebad46-d420-32d5-9562-421afe1aa4cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52eace85-dea7-3f81-8f52-bc874675e71e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a14b85a-64ff-378d-94ca-f2b8898d579a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c464f432-a21a-38c9-9715-0b5fec4d7471_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb12a4e7-85fe-3aa9-a15e-f77e278b04fc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9398f4b-c16f-326b-80eb-5a6faa3866fd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38088ea2-9428-3760-bd8f-274379d1e83f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c920f8ec-65b9-382a-9cd7-a78c754d1827_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c212e9a4-dfeb-34a2-84fb-c105dc7a4112_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d62d4315-7a83-3285-bda8-54e82a94353e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fffbff91-1b3f-3f5b-a656-131c0599f70c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_17f6f158-032d-3c10-b037-f282b1a79f8d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d401470-d968-3fd7-a1a9-06c3a4343bfb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d19d8dd9-448e-3975-9ff6-7fef8ebc866b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc4f013c-3e79-3472-93dd-cf0c85a04973_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c79195b-2c4a-3c78-be05-2ce07e711200_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_146a38c2-a516-3e02-b3a4-37e08f320004_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_71041e7c-b233-3b41-bb35-6761049a980b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fee7d3cc-af6b-3a3d-a879-78c7b98b94d9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0083d2a7-798c-322a-98b3-21fd33338172_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ec3d95d5-8a88-3aa1-803f-97b15374266a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae74bae6-dd22-3cb9-a307-884cb9c0ae53_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d2fe167-6ac6-3904-89f3-4b0a2b52f76c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d429fd73-f9b4-3d1d-92a5-951cb3cee16f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_953e8237-d544-3668-911f-2ba75f9b163f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6171d242-9a21-3d4a-96d2-a23c261f38dd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3608f5aa-bd8b-3ea0-938d-fa925a10ae52_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_babf8e65-78f2-37af-a8ff-84884b5fe488_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82b461e5-8a56-39e1-a33c-27844929d893_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82c4d969-363d-3c47-a571-826d434cc072_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_32cfa431-fc2c-3d89-9a96-96a42597a339_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a8d0b8f-4169-3dbd-8f42-ec63e9461309_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_714fe672-5301-3f4d-aec9-82e4eaaf62ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_09c257b0-44a7-389e-8336-8237fe5d9870_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0a20c317-fe77-359d-bc43-6914fdfab54f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f30f6a18-017e-3274-aeb4-b75463a7c59d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed48bd5c-96fa-3233-8b50-e7bdd85cf84a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a98ce330-3f83-325e-a2a1-edc2316c4885_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_88553305-ec8f-3012-8d44-6c512c9e0629_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_001ace35-75c9-3542-b440-1f394e002334_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e9587b4-eb37-3aec-b1cf-4d58e144a132_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5745f1d5-cd1d-3059-b037-fde48c15a51f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_af9f254c-9559-3dfc-9ca3-4c14dfbe2bb6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d254be79-0462-353c-92cc-822b1171a352_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1243f987-dea0-3077-9aeb-1cae3a07953f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea56d0c1-1e21-3faf-9839-b2657d796ee0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed73e82c-3a01-3022-90f4-e05acb0c0a9d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2d973ead-fec2-3238-a15f-a2f669598dad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c181fcee-a163-3580-8680-b41e35090817_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a410e3f7-2b86-384e-ab30-5dc3c3e6fc4f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da5c11cf-30aa-3281-b9dd-fee05d86ce62_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f973456d-e768-3b94-aee5-0f3d9aab38d6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f1d470ef-e10f-345e-bf8c-1d3451ac007d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7c4fd4df-2252-351f-9b8c-6752a43da278_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc8ce413-df07-322b-b8b7-ecb906e8696e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_68d815dd-fa1a-304d-a74d-4ae43aa22dad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1fa0f93d-5813-37bb-841d-d27916520f37_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8caa80d-ea90-3bf4-b0ba-51f2a7196ed8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c611f0c-9d5f-3e7b-b67a-afc181431a0c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_87298174-3bf9-3e57-b9ea-d5e5e3cdf598_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8e7ae3b1-2ed7-3a32-933c-2eaced253b88_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db5a6395-ef55-3ff1-ab99-b57cebda9fd7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2333b200-d29d-31fb-89db-c769077d68e5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1c2d02ab-4eba-3c4f-86b1-98ce07f4fc66_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ce404151-39c9-3da7-ba86-d8e147f3390b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_403a1621-cf08-31b8-b5a0-f13cdbf8a92f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e7bb87d0-bee3-3029-9dd3-eec64c3f18e4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8de61833-cd73-3b09-bea3-55078bcd70ed_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2ea178c5-3ae0-3a0d-980e-d9c7887edb1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_731afb9d-fb83-3258-9874-d23aa9405060_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6316aad1-c813-305d-9cdf-a6495e1c3a73_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f564f3d6-1b90-3d10-a054-bbc17707ec03_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_efb687f0-a573-3994-b8f2-1d164419973d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9efd29da-9325-33df-a620-634f54e2d138_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da8834ee-542d-30af-9068-babbfd47cc65_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1463a4a1-349d-3da9-90f1-ba9a90558a87_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_184205b7-0493-34f8-8055-ad7b486f2796_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3d25bbd5-e4ba-3e84-8a90-503ca49b52e8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aa158de3-d399-3ed9-8193-33f9f0773cf9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b6aac91-a3c7-3e2b-9cb0-e7fa53e64235_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12f2f069-2a87-3999-9e8a-f27c4a5f2ba5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c0c4c684-4481-347d-9c91-b5e08839e59a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea170861-2e65-3220-8ec5-369d5de015da_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_076d1e4c-706f-32a3-aafe-5ecdbf7b927c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d841c546-749d-339e-9130-7dd3f3a32cd7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d9c5fdf3-25b5-36b4-b4af-73e9dd6e0f17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_507909a4-fb3d-3a3b-9d10-05a0213a9ff2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_84158122-d84b-376e-bde9-7f0611146c41_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8706989-1746-3a72-96ce-a5a90aef30df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0eed1b08-9930-3d7a-941b-e873e039e4fc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ea7b7a2-4a4d-3e86-9a57-6e2c93f744ae_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46126417-1b37-3eef-bf98-53d0ceebebc2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_86667ba4-e7d6-3468-9598-10d91fa2aae2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f32137d-ee24-3ac3-ab67-5ae7c626c203_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee7a6e71-16b0-31ca-a3d0-9cd68250e8e2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e2475064-73a5-31e9-85c3-5a497c71730c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fd7ee9e3-2c69-3f78-b426-1055cf228aa0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91db922b-8dda-323e-95d9-719724f3eed8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d94050d-a0c0-3302-96c2-231dd5d04f03_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bd200fc-0d62-334c-8d8d-95e232da9b9b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5421f6e2-8ed8-3442-8027-7a34f9f0bdba_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49db8e10-5708-3264-b876-e786f80a70aa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0895967b-86d5-35f5-9844-9e3b565e7d97_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b94f19de-53a7-3fe8-ae1d-429279591cf4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0757f450-893d-3615-abed-d893f2132619_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d9432b46-95bb-3a56-8e5f-ee443a2c155e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cea756a0-7477-35e0-9a58-585ff3827895_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1c359bf6-6750-3443-9b58-7b0ddbe66833_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3e3bb0d0-fdfe-355a-a1dd-8c528ab526d0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2b021e81-44c3-3693-afe5-a2729ce21fc6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53277303-5c46-361d-a45b-dbfd54485657_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cd503b8e-da9e-3ac6-a1ad-346ad68b5d36_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dacd5594-eab6-331f-94af-fbdb9c8885c6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fcf02b38-a153-3ae6-9a4a-bd7fcc39e491_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c619fdb6-058e-3ff3-ab84-1ea7616dd05f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e4042d58-b16c-3368-a093-5166e9909e15_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_59d6cea0-3088-319a-b726-ed903edeadab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f585ddf1-87ad-3004-a301-303505433ffd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45f68b0f-d387-3d2f-a393-ac2ebd79b442_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8deb265a-dad2-38ef-8173-f890979e2f7a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_834902ad-841b-3efb-a0f8-32645a17b7ce_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a2628501-539f-38e8-84a4-c2c94ee75e3b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_849c7b23-6eb7-3bff-98f8-63fe88e5d5bb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4cdbd4ac-4693-3c78-86a6-44dd443bdc66_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_99e56dae-b7c3-36b6-96ab-ee7556aa07b9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2c38741e-12a2-3930-89f4-53b5c037cdd7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c2a7cdf-ca46-373d-8df4-ec68831fb9c2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7f1687e1-9b3c-3355-8a67-4f335b5b2879_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_67a043fa-d11b-330f-a36f-d3fc49c2293f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4b74196f-0c38-35bb-8a9d-f7ecd522b0fc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ceb8c0fa-756d-31c6-81c1-a33672c88f1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ba23d57-f113-3295-bced-13d78897ed95_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_85adc959-cda2-337c-987a-6b015ed25b52_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab643ddf-0e9d-3feb-826d-005000eea564_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_57f69e88-42cc-392e-be00-119fc00e45d2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eebc486e-9f3d-34a7-8851-d3588c895c96_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_55e754e3-0868-3f3b-bffb-4b32bca539df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7de000dc-19db-3fc1-90f4-bee2d11f23d7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aeb9c1ff-1002-3e3a-8f89-6a31baf8c28d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f761d55-53ea-3a16-b2c9-86ae508553c2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b4ece2d3-628e-3789-9f4c-5299b18913ee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23b82881-7f80-33a7-b0b7-7cf01fde23fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a44cc73d-a741-3e18-a426-4fbbf1578169_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db1003f2-0742-3c4f-becb-b5157fe69463_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_88c81479-c754-3c70-b7ac-46042b6bcc23_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db9640a5-3d2b-3898-83e7-d2cbbe4237f1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_273d7b58-33fc-35ce-922e-e0cb23977f13_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8de4e735-7c77-3269-82b7-a512232cfc5a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1da6ee14-925c-35a8-9262-5b2378f022a3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d29730dd-8343-3d13-8d61-45132ef03ba7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5238df15-d3db-3aaf-a0f2-a965d4dcac53_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_21e432e6-0c95-345c-8b10-a2ce621df9ce_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ec9dcb43-8504-3e24-b7f2-a5c3b1bee409_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_410996e5-3ffa-39fb-84a9-f3cb2a69cc88_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e4d553b-4358-398c-8f24-0097aaf996eb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1944749-1b62-3405-84c2-b560ca19d6aa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23e8fa82-885e-3d52-964a-b9751302accc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_81c54fba-a24c-3058-b873-3bd687e55048_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ea5b963-3589-3703-8e30-f15e7c545f63_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3c159983-0321-383e-9e5b-ee8cc45b62dd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_372e8280-ae24-391b-8ce3-485b2ef7cb2b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8997beb9-0137-39fc-a140-09b40e808676_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8a4e4c8-fd32-3f36-9070-5380abaaefbd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16a7a20c-be4e-377a-aff9-2f31ad54ba76_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e499c33-3985-3f57-9e4e-4d0851891ee0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e248f934-3d8a-3aba-a1f8-359ccfb5a46c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ef45d86-48af-3d7e-96b5-8c2deaec1e72_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e180279d-da8e-35e8-bc94-9ff9e6a30be3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6d0915df-d11c-3268-8589-6af7f6921a0c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5905316b-b517-3b8f-ba83-d5ad129b51e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7659a2dd-2583-3bcf-9e6f-b738449d9958_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_594c6e43-4d76-32aa-8847-bcf29bf4992f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53fe4238-8348-3322-b350-9f5ff19dc8e8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee361b19-4cd5-3167-8d20-497c70645f1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_860f53f7-5cdd-3530-a17a-300d3c8182a3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5b12f8c1-e5fa-30f7-8990-efeb41dc71ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cb9c8402-da45-37af-ac13-c02ef2ef5eb2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5d75df4-7079-34b6-919e-b12821517d4c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fff38cfa-7b69-3116-a805-bdf22b8d0cc8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3b62e137-9235-3033-8bc3-6f82f1b86a47_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_203294f8-bea2-3ee9-b236-f38f7e272129_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_733ad4dd-dacc-3f73-a276-8e1064792036_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9730475e-b669-3800-aef1-9b476d0c2e7e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ac5a2b0-02f1-39db-a5dd-a92bfccea84f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1f5b66b2-695d-36b9-b587-fd33a705d385_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e219dc57-f72a-34dc-8e87-40b0fb799256_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e2ba1e99-2e50-3454-8124-dc79a8439922_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d1954e02-e5aa-39b5-8ac8-ecc8036f79aa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bae9886e-d0ca-3339-a3fa-64eee09d1a6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e162aa7-043a-3f39-8b01-5b48d42d6db4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0dc488e4-a6a8-31d2-8325-3a00b1fedf47_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dac5f843-3430-3c80-8bb0-f8794db7c66b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d419fef5-444f-3478-a3bb-ce17004d4c2d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_528d0a8d-23c7-30d0-bbeb-7a81d0f5df79_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eed73f95-3524-3af9-8e1c-3dbb330c711a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f61f49b2-b85d-30b3-91c5-10ce1bf9d752_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a289bf1a-9d5b-3622-b4b6-587fe67104e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_beb22f95-2a2f-3bbc-892f-ab30605cc9f9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5323b589-52f2-3623-98c5-3d9a4f50734a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12c74576-a4aa-387e-8536-d20372204c59_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d2989f8-ca76-330c-a5fb-ef1292873887_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e3affc9-feb8-3326-8e64-0e7f9babbb98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90bc75fe-3e90-377b-9bee-de3b30373cf5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d74a62e-b9aa-3aad-ab3f-b64f3caf1854_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b95d228b-586f-3060-b3d0-dca7f75361dc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_229065b8-b0ab-365a-81e6-de93dec5e706_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_709ad331-cdf7-3e56-b237-741e176179e4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f0cdc697-a0e7-34e0-81cc-3740882c5ebc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_59ca3cf7-b631-319f-83d2-c727f1205763_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d2be93fd-63eb-3f15-a87e-db0499143573_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c801630-a9e8-3848-8520-a828b273d316_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_380ec176-13ca-3df3-9619-9a48fb8e1bae_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_03b4cf8a-3a9c-3262-a318-be89af033483_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d5073152-9c38-3bbe-b915-bb14cb381848_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1156a4f8-17b9-37bf-a36f-a26c19226192_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_81e6246c-2475-38ac-8d18-8211cc33d0f9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd07db1e-e742-348a-9af0-42abf897b27e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f247529d-a799-31c7-9ba7-1d6e4b8c0611_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23d192f4-cd5e-37fc-9150-fb7e7384014b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f8a3eb64-40d2-36d2-a5dd-e2658b3d1282_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_43a2f4b1-3712-31b4-8176-18cceba2750f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc45ddb1-029d-39a1-9f43-ad6a8ec41ae1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6144ecbe-59c9-3927-9e6c-4b9ff55711b4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f9b37d0-d8fe-3c73-9967-666447a88aad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e03e90eb-ecef-37f5-b1c2-202398605bf5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b7a36475-598e-3c78-ba00-745680293772_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_17ce3089-8ade-3510-9eeb-8a19d07eed2f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d6857b5b-1d4d-33a3-8baf-5c2346b3b11f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d151216-7557-3bc5-9b63-13e5bd495fdf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6dd31c25-c7fa-3338-b0af-f9bc5e30f330_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d36d6486-be77-3f3b-8d91-034eaffcb666_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c8e4952-3578-3899-bcd7-04938eaf8509_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d2ff7c8a-7212-30c7-a119-09e9634c24d8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5645a048-3baa-39cb-a1ae-e0c4b5922b49_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7fdd7812-4679-3b90-9e84-93e68e749b23_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b83c5b5-cab4-3b8c-b60e-7a53ef125808_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f3f0cbc2-4a80-3619-8bd6-c9e1f61ddf06_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_625b8280-f8c0-31d6-a46a-cf42b69bf902_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_046cf0dc-d1b0-3641-9e86-6cb7246849d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_41205fa3-74c4-3f7d-988b-84718f0c9a4d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1675a7d6-3112-3079-8d2f-36c30eb819dc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4f4a6646-8561-311b-8eec-bf573801bfd1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e0abc0e6-7a19-33a0-a337-768d7189c527_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_97b729db-9e93-361f-9ed6-7ae6083cc32b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aad9c45d-fed7-3662-b8f3-4831d8052bdf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd1e3852-6e5b-3949-9053-d9ca7c558db3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc1da22b-c858-333e-8417-0aa1e1ca9a1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_304a3b92-cecc-3673-a555-88f9372e8034_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ecce759f-d406-38e0-8940-e2802c77f2b7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7041f50a-817d-3cb7-9ff5-ca7d7af95ad1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1b4d5dd1-9da0-34cf-ba1f-8727eb69fcb2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_04b2285f-e5c5-3004-9387-3dd584450dfc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1c8103d8-1c93-366d-8eb3-b94cadb19635_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1dbde513-debf-3cfe-96d0-50c7f9206ff3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_89071f40-a234-3a92-a41c-21a0aebd57e6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b399825a-1e92-3c3c-b18e-00d8bdd03737_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5583b738-07b6-3ae6-b110-a07530f75bfe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49181a75-bbdd-3ae6-98f6-02869036f39d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d84a2c78-5859-3360-946d-05e9a8d095cb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_57b7ff48-f542-3bb3-bf22-a539de902237_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffa15c3a-fb95-38ad-aa03-93bffe97adb4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_96296f04-6e10-3a1f-bc26-66dcc41ff18b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_481ae19c-bbc2-3f28-b47e-96a8bccc6faf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c36b22fa-7db7-3082-883a-6e0958c7d6c9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d71a338-fbcd-330e-b328-1de48eb64392_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_491331e7-36c8-3990-9e3a-2b38a70185cc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7dc5c610-b309-3cd0-bc86-943b9ebcba99_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6eefca33-d902-3069-9075-3be2d85e6fed_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5bfa862c-0512-3c58-8130-2dc110c5a67b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2783835b-95ba-37a8-bccd-a864536e0a40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3bb2955d-ca66-30c1-b04a-202fa2645361_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_df077328-cced-3079-a266-9ec2d2fd4543_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cb8368ed-b839-3345-849e-62a585949e6c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3aff8353-7bd7-3e48-a063-7a10b7b29f98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_31cde6ae-0043-38d6-863e-064100522155_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d028037f-2e0d-336b-8c5e-b4e4eb90b9e0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd061d25-2229-3756-ba26-3067e276ad19_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_77a4a8d9-2d79-3ee0-89b9-e1162f6fb2a4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_abfca739-22bf-3021-a47c-43c40f530d42_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2feab97-49c2-3917-82c8-5f94c9a216be_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ace94704-a6de-38b1-9b55-0b08d5fcc7cb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3a49549-6e88-3d83-8248-f60bdeec6d33_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7a3a23d5-4c0c-36e1-ab39-db1689a0de0b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6136081-3b1c-3f8a-995d-e6f52a4094ae_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5d574e93-de30-3088-bf01-7e1e26eebcd8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fb3a2982-7a6d-3e60-a15c-91f551b5e4f4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8fdd133-8a31-37b1-883d-1c64d086888f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de5492f9-e0d1-3d55-bf9e-786ebbb09737_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1add14bc-8137-3b66-a149-90f1cc4d217f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2246ad6d-e0a5-3203-bb30-de78026dcfab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4256cce3-9c1a-3e8e-a756-44d90448ed5e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7eab86fb-d9ce-3313-ae38-29b78924cb3d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee3cbae3-3377-316e-8290-f051c1c7540d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ab2bbdc-5b33-3ac0-bdc8-c0410568c9ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_67007dfb-58ee-3011-b5a3-d9e24a89d4d9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6ea5e5bc-d990-3009-aca5-35263d79c50a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1af4274-afb2-3070-9699-a8afd98e82ab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3db7ff76-590d-3540-a86d-84c45255b332_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0113ed5f-0a42-3c34-b62d-8535c2a59d06_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0e06cf6d-e557-3e26-a599-dc603313097a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_009f17cc-f29e-3948-8dfe-aa5b4de67b57_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07b41463-4f04-33d6-888d-b4e36112fd01_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ad1fe30-fa01-3084-b0b6-c08dda919db3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a115be3-1180-399a-af17-f03e670cdf2f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eb1b36f0-d5ed-3a7c-9d12-b1230527bae5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_05faa8a4-5ff4-3e44-924f-f742cc16273c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e00ad7c-818b-35ee-bc45-587c223c42c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da1904d0-dedd-3968-ab4c-d3a1865ce3e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37033b95-8a5b-3f98-81c8-b4c5e7287434_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_667cda6c-ae6a-3e32-82d2-4a523d0b0184_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_72e774f7-9ca8-302e-9f14-a8caacc4fd3b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7674ce66-1a02-3d51-a1be-591fed1a44bb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3fc11af-5ff9-3675-86b0-bac5f58215b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ff10693d-35e5-3093-89fc-d9be32f5d8e6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ac81c9b-392b-3870-ba48-8256a9f354f2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d39a97f7-9060-33b2-a29b-a12c951a9b30_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c84cd202-1db5-3fed-a747-8caeaeacc3e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_01dc75b5-9167-367b-b34a-7eb60123a23d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6ca09a4b-af4e-30f7-97d8-5e5fd5af43fd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_39b220d8-9e4b-3b16-9a83-ae16aa1f25fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9c9659d1-0754-3e07-93cc-d13615fd1c7b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d83efbf-c383-38e8-86c5-c46dfe0a997c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e13dc471-b1e3-3d7d-857d-947237e898e3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_911f29e5-e3c7-34e5-90c7-844f6df3f5fd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b95e1436-b026-34c2-8e07-1152199d9b92_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3cfee524-dde6-3730-b5fd-d8758e31293e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e76ea316-3fbf-3825-8640-93cb23d84b6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b8430537-331b-3c29-8127-7b939dfac03a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8b1902df-649f-35c0-88d6-b674f3f4c72d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8be62e5f-5ecb-33bc-9363-45d4e54f1f6c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3145127-3fcd-32b1-89f1-03282086f47a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5885df05-adae-37eb-9639-dfd0e9eded4a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0e05c27e-a19f-313d-8f88-554c0459405b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5b7585d7-293e-3536-b728-9243e5c81b49_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e06c3602-2093-3235-92f6-f5c3f6afc263_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9ecaa20-a17e-3af2-b41c-eefe3e6c2183_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fb0aff0e-b347-3e11-8a25-762d7791e634_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da01b76d-70e8-398f-8c0e-206d308590bd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ce9f976-4b6c-3faa-9ce8-434cecde3b70_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c27b18eb-e96d-3e96-a9c4-115987939530_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc915db9-8d00-39c1-9942-215931d790a4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9837ab32-b93c-3c8a-92da-06c6f66a00d6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91b97555-8c41-3cac-bb70-cdc858cbee35_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46296725-ed81-3f36-bfb6-3c1e006722de_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d2bfbc3-e46d-3e3d-89fb-e137d1d437a7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_43065314-5de3-3259-af52-9378e813ecad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8771f5fe-31f3-3c76-95d4-bccef8c1336f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e2fd6419-c77d-3520-8fff-7ad8a56e0ba8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e15112b-786a-310b-9fd4-55a4ef4286fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be90b3bd-a678-34e4-9ebe-e77032837477_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_66e8af22-a544-3a7b-a3de-6f46071776ac_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9460d9fa-cc80-3143-b9db-651f20e60c62_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5096ab28-31ec-3558-9c56-8590bea8a172_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_02206bbd-6f62-39f1-b3de-4c217e017bdc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6dbb7a91-64de-3be2-8e5f-2e715c8d7e9a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b4f74a7c-bdac-3885-b286-27d438ceb4ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1c66f082-e081-3149-b083-3a8abcee9796_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0481817b-d363-313c-8072-394747899cb8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1def39cf-a166-3ae8-acee-33fccbe5dfbd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_58180aa3-db32-3340-8a80-f8f2c951cb8a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b393130d-7edb-3438-b8db-7e5c32d976cc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e96122d-aa09-3e16-ac7e-d14678ce8d73_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d822c5b-2e8b-3542-a8f5-30980e47721f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_629e4959-459c-34f1-b730-6cd5dc1f48e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd060b80-6f4f-34c8-a8ec-20e877110434_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa17282d-ad2e-3d90-95d7-b07fe96e8358_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2379ff91-e48b-3561-91a0-43a20fb159a2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c8476c26-39db-3e3b-ae28-e0d0548f37d1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffe9a8ae-5a91-361e-ae0b-add549c6700c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ebfe2146-28c5-3484-8142-3db2af53df16_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6cd5ae3e-f988-3699-b5cc-985fef84f537_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80bae5ca-3982-3bce-9f3a-be31553ebb75_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_98f96b52-8026-3b18-86b0-5cb6ad238a17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c793f601-151b-3740-ac0a-f613cd48f085_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_365f6c50-072f-30d4-a9ca-e018c951bbaf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_620949b7-9c76-3ac5-b42b-4792ff19e3d9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc9d5025-6890-3b7e-a189-2448303af2fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9b5a2ceb-8bc7-3e78-becf-7595acb73912_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6559d7e3-6fca-3c60-ab20-99c1a2826dbf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6393d075-eb89-3f56-a50d-9744340afcc2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b989124-b6a3-3a9a-bc46-5c387a785058_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_28a3d56e-4b98-352c-8e26-6cdd087f4cea_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_832bf82d-d145-363d-a747-e76850bc69b0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f20fcd36-8039-35b2-9346-8ef7afc9f8e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ff8f052c-9f8b-32d8-8a78-672fa0bedaa0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83b04c4d-d5de-3b6e-9cce-285d110bbd71_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_19537822-68e6-3b02-87a2-e19f9a06432d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_757296b6-4045-32a8-854d-ef551cca2f4e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab86fe2f-54b7-33ef-b046-13dbd91bc8d3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e0c2839-f5fa-340b-888b-b7bb697c20cd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8dcfc20-8698-3ece-a045-122bdffa52e2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_27ee2e80-7b3d-3273-92c6-bc818f522ca5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_653121b2-0891-36ea-9a99-111f3652c31d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ced6225c-5a56-3a42-981a-f90bbdf0fc74_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d6d36d37-061a-33ba-aa9a-05d885e0588d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_597aa2df-5173-3f98-b47d-07f3cc02f0e6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d37c5546-3b87-3426-9011-15bd69eaa303_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bac05d80-c0d8-37fc-af96-9482a72c9d90_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_158338f5-caee-326c-8ff6-34f8c9445e41_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3dc65494-2e36-30ad-ac22-e73aff80cbfa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4935c4b0-0585-33d7-81d1-5623d384ec0c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd1b3593-02c6-3c9e-a901-cd2cc9d2f16d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_518692ea-fbe0-3874-8480-0787f8c385d0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26078341-51fe-32bf-bd29-66d05b862ae0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_08725500-8023-389c-b0c2-94a603ef0f17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fe198dee-f250-388c-b603-dab76ba2816b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c287e0aa-54ed-35fb-84cb-dd7a88ec369f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eaac6c1e-7de6-3cf6-9759-6f648031329e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5ccf0253-a998-3d6c-80b0-f4df9c4cf1f1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1658d7c0-1320-3b23-8cb6-cd6b779b5488_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_95f696d5-8c9f-30c0-8d33-3006d8e8c3d3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e9d0cf86-9529-385a-8c97-58c413eed518_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8fe49242-4e30-3d48-bde4-66a2cd80ee61_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_26473a41-c0a6-3b98-a4d4-edca5bb01bd4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82738b32-a620-30f1-8c53-140c12534346_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a9d0907d-d747-3b20-9840-9a899686f37c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba386059-a994-3f1b-99b8-86d0600878dc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_785ce2ac-8d5e-31eb-a241-95732da9e5e3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d2bf7f3b-c973-3d2a-9d64-0b9c5adb3d05_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d93cb5a-6f35-3887-bc3f-25af29610a1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d84553ff-7f51-370c-bf2c-36ff18998dba_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_18b383e4-d621-3630-9384-6d34bb5d6098_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0397d761-8e36-35f2-826f-98c622ad8466_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a78118b0-b4fa-3f78-a940-30f86ea36901_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3d5c419-3863-3fb3-a233-c819afde6dcf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_72b4461c-4d77-3bfb-a459-95007323167f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d9a7d92-ca9e-34c0-8b79-993d022112d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffe2779d-833b-33fa-9488-35e1aa3e0247_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cdc12b3d-605c-34a1-89de-758a54e37cce_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00e84e34-026d-3e9e-b428-311e19f54479_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8c72a7c-1eb4-350a-bfc0-7ab88a88c85d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7fa222b1-e5d0-3d05-9a74-a41f05b61719_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3226f352-04f5-349a-9d63-d3df99457f18_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ba830dd-6964-397f-8309-92c3b84e5068_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a7cc298-fb27-357b-8375-9eaf8f1aa003_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0c90471-8ad8-315b-acdb-522db4d460b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fcae4165-de06-3683-aaf0-6d05d2e4d5be_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2af46e93-1230-3e9a-b349-e69c82a7d743_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56ea5e0b-a546-32f0-9209-eeade2b75662_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_60c29bd5-1744-3295-8587-029061272d8c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45ae9c54-7cfa-3a79-99d4-c852ce50e62e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a0919e8c-74bc-38a7-b1bd-8adee3fe5b51_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_33ebcc30-3401-37ca-80ed-3692daf4c8ee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ca088ca-db04-3f01-b177-38005748bb26_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae6f9f48-0ef3-364f-a099-657b89168798_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1901038c-5118-3697-a184-a479d540b679_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9da53ef6-c682-3eff-a25b-34f8d49576fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ca72d6b-edcf-35c3-ad0f-e14aedb578a2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_239a1bf0-c84d-3a7f-9a52-239fa70e11b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_81727ab8-6822-3250-b116-0562917fe33f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38f0331f-cb78-3855-86a7-9f075e145dc0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ea59eaf-43c4-33df-93fa-d11b965cc1e8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e88cc05e-00c7-3079-97fc-6b701260b58f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6820d458-4413-357f-938d-859acc4bb02d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_64651953-c1b7-35c8-9240-9201c335f335_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_abe29f3d-6a6a-3dfb-b3c6-5ae52656e484_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_917c6543-e449-3e11-86aa-36f43aaaf26f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_392d4fce-84ae-341e-8d2b-7777515b1d98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52573a97-89f5-33c2-8229-ca87f3aea6d6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_15cb4f5e-585f-34ce-adaa-d2f5467e0eb6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46b841d9-96aa-3c7d-9e1f-60b247b27000_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1208008e-3585-370b-a1ae-d9392097c527_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1e2f7888-bee4-392f-baf7-0b4ab37a16e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_80574104-868a-3929-878a-7bae94aa9b56_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6bdc953f-afd2-3f0a-9c47-8800b3d0ff1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3151c7ec-cc53-3fcf-80c2-ce6959c2d565_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef8a343e-d9f3-395d-a577-6eb672083a34_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d270e20-5cbe-3269-871b-11bfb85c66db_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a22bb05f-ae3e-3101-8668-742feba8a79a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b2713402-38c4-3c74-8073-4ed18b4ef7c8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_651d72a9-076b-3ed4-955a-0337d2bc76f6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffab2a40-8c03-3d23-bb43-d963487090df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d0a4032-7454-3124-832a-366e138247e5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8e9f93cd-ae34-32eb-ad12-6552f51f6106_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0a096c90-6fea-3764-a404-e38d00495908_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ebf9957e-c76f-3993-a15d-402d4eeeb78b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ce7250d-7179-3cf8-9a8c-5e1d9050f4f4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9c36a800-6c41-383d-9e37-4619aa8979a6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d35cb8a6-acc5-3082-99d5-0b4aeed261c0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be06e44b-cd64-380c-9743-530dd49d0a63_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc9df0b5-ad32-36ce-986c-1b61af19e958_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ad7be270-b728-3bfd-b6f2-987359d216c6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1b9419e5-cce6-36ac-8a25-4cd6f545dfdf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aa713813-58a4-3b3a-94fa-45b0b00ba74c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7f27e8e9-11dd-392e-b130-3bb06bc089ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa73531a-6eb0-3d75-8375-b05a3f1d9a04_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d9ef43c0-6965-3a00-bad2-4372fb2826fe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_54bab469-6b18-388a-a45f-dfbe9a6baf67_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83711aac-c044-311e-bad1-bf347490fa5e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_293215e0-3eac-33b6-b95b-a058fcabca08_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d2312865-278f-3341-8a55-96b7808e3400_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_370bacb4-bf55-3ea0-849f-3dc78c004057_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f59a764-f7b8-3101-a959-efea36fb29ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a22ab9eb-0191-365f-bb4d-f9c107df5137_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_339635e7-255c-32eb-a9e5-2fe306a2b04c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0876854b-1b70-3a72-a09c-1fc7091cd22c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f336356b-b33d-38ad-9380-00853e05cc50_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_608f1e5b-812e-3acd-a6f1-7b6a37cbdd12_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e7aba6f1-f376-3087-ad83-ff3e71a4a837_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d78279b9-a243-3f94-9e71-062e7335879d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a5536a8a-5dec-34c7-8cb4-f240caf9db2f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f86a3274-64ee-3b31-b3d6-a0c43d39b42b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d40d974-a0a1-3290-8259-acc29e3c0a29_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6cc4929e-57a3-30bf-9b75-8fca9ea2d4d0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2245f3bd-b799-3a5a-b04b-5c9f7bf7b309_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9f154d51-603d-3f64-8da1-1a45dd732b1d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_160f1aaf-d8b5-3686-8334-320927ad7ead_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_88d13438-b3e3-3f48-8557-02408a57212e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4afe60b0-1b41-3a2f-9e0a-76d07de0e0c1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d8f745d9-0917-3860-9583-19fba0c40dc3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_43e0be0f-cb6c-3106-87be-95890ddde1c8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a70fd6c6-2a39-384c-8475-e2cc148b87a8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bbb37ea1-d25b-3fac-8629-794f91571453_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7cfc1b63-030d-37ed-8789-4ca4a79c202f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7a7e689-dee7-317a-8f0c-09d180c40d75_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f6bdf180-55b1-3f73-b942-4271c6c61cdb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d1e1f11-59d7-3f31-909d-54cdc7e47d41_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3efe40e6-8ae9-3aac-98c9-9c0ae2bf785d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cc24a754-f540-3085-a262-ae48b449dabe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4516e749-f116-34f2-a8ac-7a54bf20dc40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2a0ea37-e3d7-3ae2-b4f0-c45a6dd44137_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7cf87c42-86f4-3003-8918-542a98eb29a4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_21b1c47f-a4e7-3113-81dd-5e5befee8732_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_687b3b14-7e4f-3cae-be55-ee11b37e50a3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d5ca39c9-30b7-3060-aa20-f05b48e4407c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38d96e12-3834-39e9-b8cc-bc2336041749_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0d71e764-8733-3a67-a0fd-ac49c1af8912_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_57aa8036-46b3-39b0-854f-d9ab446cbbdd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_29709643-3087-3ca6-9246-94df55b699b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a866540c-922d-30d4-9802-2fe9ac224cd3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b98dcac5-765e-306e-8fe1-e3a4d2bd2796_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2da39503-de9b-3cb6-9fe8-3dc917891633_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ab3a46b-8e01-390f-845b-d41f2bf74e34_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e08fb129-7e12-3eb2-8a72-79c0187cea18_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_41b07434-02f2-3f16-a0b0-7c997b99df52_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_19f902ef-ef9d-3e2c-b0cb-b6929fa4feea_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_95b90a54-7d86-3e23-bfe9-4a22427d3b42_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_24397ca3-a9e8-3b2c-b727-b26b3081ea0b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c5121742-2a64-3182-abe4-cb270954fa94_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45d85d2c-4a07-348b-a213-7189d945694a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f9d7f184-4af6-3e55-a2b0-75b44517e410_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a23e6dba-2f87-3a19-9052-3298c600ea79_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a9b06a4-cacd-3587-adb3-6861b8d845bb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d9b62fa2-2ffc-3252-a679-dab93a263335_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_846137c8-0032-3589-a711-704929055a65_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db6f6c9f-2797-36d6-b820-a0a0cece462c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_acb76ec0-0d92-3215-b66e-4918bed3a20d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9b3d2f8b-5b98-340c-ab3a-98eed8247a23_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e05b22b-2da2-31d7-ad39-55905e4acf20_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cbfe7c61-e044-342e-b4e9-50fe430a661e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0d886136-ff62-344f-833e-d6a1fd0f47dc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2dd17ab2-9def-3eab-b2c5-5b0aea1084a7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ce3c429-eeae-3130-8dd0-668497aec53f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3449044-812c-39ed-ae9f-f0272058987f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae0adf39-894e-3f24-bf5a-5bb4705202bb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e75813fe-b3f4-35e8-93b8-a5f22569800c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ce508837-f8eb-3f2f-833a-ac9a3b8b10cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_68807928-7884-3dcc-8882-ee28667245ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37690442-9945-390c-b70c-4cf0843a3f95_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fe5464c2-5b0f-3731-90fb-53fec4c059b9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cd74281b-3f8c-35dd-80cc-b15ea83aef43_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_55418b7a-b02c-3065-a726-da37c4419fc2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3fce7173-04e4-33e5-96e3-8d414f7a2ddc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_34e61c74-728f-3a58-a093-3ac9ae3deb81_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ff89502-768e-37c6-85af-80c565e2801d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee50ad10-2121-3392-80f0-41177da02694_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8eba7da-f24b-3830-aab9-c55b603c2cd6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0ba5b594-a5d5-37ac-8cc7-646ab7320c24_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46ec2991-9861-3a57-b3de-05df064af2de_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c59b5897-c09c-3c4e-ab57-5c0a480d98e2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_509b5f13-a6bd-37d2-b98f-5c34f0640395_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f40e2674-3f43-3dda-b173-188f30c47c35_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_368ffce2-a129-3f58-94b0-ee35ad0f011c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a6a4a634-317f-3909-aa9b-ec3605db0470_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f113d1da-c019-3860-becc-5ff8d01f3943_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ee4de0e5-4b84-3fcf-b363-ed45f576e6ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da20c386-5833-375d-adf8-26cc16ee68ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12f26ffd-fd3e-3e0f-bf38-93aa973f2e72_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c31638f4-e47c-3e19-a83e-b4d907624b1c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_14d6252c-02e2-3946-933f-600e44ad494f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db893f5b-051c-30e6-b6ed-208e2f8674e4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5cbbe1fb-8d5f-3117-8b23-e03f052c9fda_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37417f06-7639-3ce7-b93e-811d5f1a8911_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ba39a36b-6cd1-3e68-83cf-f8461d87ee19_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6864721f-3d2a-349c-8ed9-f65746104f40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38a23d46-2e0a-3df0-a1be-7ca74ee8964a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56d16349-c348-3ebf-825f-c027315dcfe1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc9657ff-88db-36ab-baef-8f6e663c7d56_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_91c4edf2-a7d2-3292-9c82-4a6e88d1da82_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52f9c87d-f9d5-3451-a03f-711d0f4b1cf4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dff54114-2858-3230-bb23-4321ac648335_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_209331df-0072-3eeb-88b8-40855a746974_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ffa6d6d8-b6c0-3c6b-93db-de53371e8feb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2496f49e-3938-350c-b6b5-068c280236d7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00412bbe-6e7b-3898-b260-3e5bf9466404_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e38cd30a-6eb5-3717-bdf4-7b8566edaa7a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1a406ff0-d5c2-31f5-b788-88bedc3cd58d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_02373678-d9f3-3a96-8c5c-eab8587ebc43_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0c3a99cf-5828-346d-91b8-d6c1b63e88f9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2049acb8-a33c-394a-8dc6-866a733c9ee9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f1ddd6c6-746f-30f6-a4df-d6f82dab9aec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0f883531-9ab6-3788-98e8-9c8562bc8234_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_312158aa-9121-3d06-bdb8-81ff53b270ee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b431e36a-6b56-3f06-84ea-4111f27c494b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37f003f2-5e0d-32d3-a776-bc1d289649b1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00a9c516-f8cc-3233-9edf-8fadf30a5aff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56fc09bb-5231-3b81-af8c-52ca76349434_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1b737501-fd6c-32e6-8283-2e488565a125_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_072d9e94-7894-3239-86bb-cda1ed4d2235_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ef7638f-13b6-380a-9cc1-52460b5b2eb9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7c04d2af-7e40-3d3a-b18c-2b7e1d6bc2d1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4ed9cb4f-3358-37fd-96f2-722c4b623f60_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3addc8dc-ce0d-3864-a9ed-4423aa512fc9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0da6bc94-18e5-3c72-8f01-26190aa75c25_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1fac8112-b969-3a6d-90d6-dfe12deb5d6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d3b5bcf2-62d3-3ade-8e8f-a3f91106b0ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d0799b4-aa30-342e-a35b-f5d94bde2097_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61cdbd3e-10d4-3cee-b00e-ade44af7a5b9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e22d3bd-3ba7-397d-a543-822c5200ad54_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e4e93da-0d30-327e-80f3-e63f4bd080d5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_507bab23-7cbc-30dc-acf6-ce102a04a1a5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52306e3c-608c-3cda-83fc-f1210e0de955_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_70f194a7-1b21-3b06-bb28-856c24500feb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fff20a3c-1d2d-3856-908f-e715d11532e3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d2c441c2-33ff-327c-865d-7e594f091c0d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_086d7cc9-bb43-3b88-b27f-4ac2e23e767c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4aecd205-ab6c-370d-baef-d894b85992d5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c52340ac-f4cf-39f8-8b07-d0389eb4c2a2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f41ac58e-bc2a-3af2-8c41-9251e57a15e8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f8fc9d61-2519-36b4-bdc8-3fedda283e02_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_234d1e37-7879-34ba-9308-2d0c86d84c5b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed615059-8887-3a30-8487-986f30d436b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a2b7f278-a380-3f91-aa29-5eee1a4f0a28_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6c4cef6a-f412-3ede-98af-ea485c1df042_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f643b6a1-1442-3a19-bf34-c2c761545610_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5dc3082a-f8b7-381c-a391-22555e59045c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ce4fd98-1c15-332d-94c2-00f1bf6319c3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b45495e7-f31b-3db8-9d09-94dc749a15a7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6dbcf5eb-0048-3133-ab7e-afce3f36b416_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e40c6e37-7cd1-3b18-abfb-3ce2b236e5df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53e469b4-f641-3423-a256-c54df4101ebe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5fa6b52c-a502-31df-aca8-59a2f347ac81_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5cc5048e-3721-3bf7-96ee-78b34cfc09c1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fd07e666-6455-3c37-8ffa-8099fe6fd70b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e8233ee-076d-3156-af6c-a9e745bb26f9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_13088ab7-89a1-3bf7-a49c-3dbad49b570a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bd17b405-4280-3573-a1b4-22d1692b55d9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_31510cea-ba91-319b-b5f1-1c10f4489a49_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_967d232c-bb63-3692-b4d4-9591b4779ac2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_94338f29-505e-3c02-bb6e-c5ee7217ad11_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ff983e3a-cc32-39fb-b5c4-2a3af5b05f62_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9af56ec8-13c0-304e-856c-8e921631dee7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a72cec31-a39f-3a75-93a3-fd631b75564f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f06c9f67-bad7-354f-a7eb-f2544bca4dc4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_957e512e-333f-3df3-b014-20bfc5000cb2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_29c5a431-7eca-3253-97fb-e0d0aca04bec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e2dad8a5-426f-3b4f-96ff-1ea05cbcd098_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_99433e8c-a026-3dcc-9f68-2593dfbc4692_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_586bbaeb-c398-3510-aaef-d16ad3a1c3d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_83516287-4824-37ab-bf97-d7d0cbc1b935_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d49bb795-7841-31df-9a0f-d7b435c3ae27_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_699f1d02-3489-3206-927d-deea20720831_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6de35046-93da-3bdf-8a12-bfd59e08e92d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_979b8d90-1ee5-3b24-b860-d41e0d04b318_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_39966eec-dca2-3970-84da-a32871b53497_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37258c56-11e6-3de4-9327-53d2548130a0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4a322989-d973-3620-b4ad-eee7a8323823_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d4ae0e7-189d-3430-a91c-14ca8d541952_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_56572a5c-068c-3b83-b924-e1ef076e10cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d2cd2a8-b55f-300a-8311-61097aba2d94_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23642dea-f69f-3063-b104-07d274a4b4ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5277e46-c0cc-3d80-a113-d15c0a4af9f3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_013062bf-c3fd-3d79-8bf9-e5c7f7c1071f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c2e20cfe-717d-3893-be27-f3e38f1db20d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_27c6e205-686a-3a2e-8a6b-4eecace5b796_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3e06cc13-f66d-3f82-bc72-0e1fb5692dbe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_521b0d29-0224-3dbf-9c37-5079db05f8f4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0d6afb97-9591-3837-a64d-9cba278b3871_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_238f17c7-3809-33d9-8b86-8b7776e98f72_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6cbc2764-2315-3a58-a3dd-9c9fa7a712fe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b2b3892-fc5a-3316-8374-956617839ddd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8c6f799-25fd-3518-9d7a-47df4287f775_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e5b9f85e-8004-34e3-856d-0cc8db72271d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_25a0113e-5e1e-392d-81df-ee289e17e878_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b20b2135-5c09-3975-b176-ea2d0f83fa70_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7b845a71-43dd-31da-9dcb-5ccd6dbf69c7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_470c9eff-8384-3c38-8452-508b37267a0e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_193fd6be-6446-3088-a7fa-bf79ae739687_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cf3e574d-73ca-3287-bb42-766b0a98d079_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef504192-471f-38ee-87d0-7ddea47d0b6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e09dcc6-abb9-3bbe-ac24-d4ea8111823e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dc3bb4b8-6f77-3a68-ae41-f652bab563ba_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_06bdb59e-a058-36d0-b1ea-cee6e9f926b8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_12833ef9-f07a-3026-99cc-f1352803f568_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e270a15c-741f-3a8c-95b0-3910188bad21_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_866d60f8-b685-3400-8ec4-913db0544b17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_55cedbc8-97e6-30bc-bbd9-bb04942e9744_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dddc79e9-0a1c-35f5-8fbb-e26345e3cf45_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ce6f471f-65d4-3bcb-8586-71a3534b063e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_52b504be-e8ed-3247-96a5-b7d6750668bc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5c9f61ed-5608-32d7-a650-b0fba6ebd31f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dce44afe-1fca-3604-b4b7-f0528e3ebe2b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_546d1604-6e85-3b9f-9e0a-8aee7300af6e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a4841324-a2ff-3654-8287-a29c0bc37b29_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c03e2ec3-6dc7-3cff-9a94-ead7881b891d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fa6e0e27-2806-3e68-b9b0-87b5c33b9cab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e91b73fa-1e82-3f1c-81d9-cf760242185c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_154cac42-f94a-3c01-b9e1-579c231a8aed_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_01327d78-4344-3207-850e-87ba12921566_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_477966ff-6614-3870-b811-beb1b5d02fa2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3aa62e15-4340-3386-bdd7-6dc0e513040c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a4959be3-89dc-3609-a782-43291452f3e3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e8c9eac-b2fe-3be1-8e00-8c76bc589c32_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c304dd58-d34e-3a16-8d41-1202bcef4922_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c5ecc89b-9eb1-3c57-82bd-c0e4c8e0c082_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ce1ab9c0-57b5-3111-867d-781eed0607b0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f31d2f53-c569-3c58-ae4b-bbbb45d69c22_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d176921-6ccc-310b-a2a8-c815eae5535d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8a75d1d4-85a1-3e00-8154-b6964df174a5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d6510cf-24da-32d6-96a9-31fc2d00d022_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2ba262c5-0fcd-390a-b7d0-805babb3e312_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4150b184-4a42-3885-8a6c-72fba2286816_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_543f692b-85c7-34ed-a9ef-608fc08dbac9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_85454c39-378b-362e-860a-598b9400252b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb961e8c-1b91-3e7b-b91e-6e69c91d8eb4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_21ac11ae-a183-3c07-9558-83e8561c77de_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a5127025-bf0a-348b-87a6-5d50fefe28e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_efe9d394-1186-3364-b73f-5c9d49497ea7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_66f75c32-8348-3506-b175-50ce4182f6ee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a34796c6-7985-3e68-9e26-fc608d6f6e33_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6348fd2a-a5d2-3c88-9e4f-7cc538325ddf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1b49076d-31ff-34d1-8543-676b6f92168f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4e914694-96b6-3025-a2bc-e1c07e8634ce_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fc3484b2-231d-3678-a600-3e45aba74ca7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_94dc9f12-3761-3dc3-a38c-9da648b937b1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7e8b2096-1de5-3f15-a5ec-4b642eaf68c1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_84437e05-0f74-39f2-844e-7fb4df88edaf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e972027f-2721-3f83-8a7c-9194b194d7fe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_908666a7-59f2-3421-a379-4c45029239c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a6b278a6-6c0a-38ea-bcae-5c8854099aa0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_82766433-42b9-3baa-9433-27313c3e4ae8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c6f661ac-c788-3448-b276-8ac848df3c93_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7c18bb6d-e7b0-3aa3-8d39-3073cf6f27c2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_798562cd-dcf6-3b79-9fd9-704d640398d2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e90fd4e5-c885-3176-9732-de53d585fc4c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90f37d7d-38e7-3ae7-bc93-7c2fb4ffd494_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b479db3-f0f4-3b0f-9c64-9eae8971a499_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_163cb96b-a30e-382c-8e96-64a3b47c42b0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bb89c206-d548-3a83-9034-be39f0986a40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d0109995-84b7-3194-9073-1946d34af535_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c968f220-b8da-3872-a344-3a3d095bf62c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d19adf0-1bbb-3548-815b-6b1dc935d6a0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23210ac7-1eab-3c8e-99be-6969b3d037e5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c8982b9-3441-3de5-b9cf-04ae63592e6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0d2bf716-d25f-3eef-bfeb-82a5344c85ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_75d7d284-02eb-3399-ab7f-4dd57d65c203_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d9702554-f42a-385b-90c5-fed5783bdc9f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_be396531-28c9-3328-a12c-ca0032a02987_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07c15778-ca5a-36dc-939b-61b1c77039ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f71b2bdd-85e9-3129-90a7-1941cc431a4b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d79ef592-5ad3-3ca6-8e09-c5ad1f0122d1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ab0dfed7-7dc6-389c-b5d5-187b6142f57a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a088f269-4924-3d21-b93c-ac154af3053d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c5d21efd-e04d-30cb-b86e-b51aa2c52c12_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1271ff69-8403-33e7-be64-5d45ff8daf0f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a185fa8-e532-341f-b91d-cd4ce7538bf0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b38f3961-58bb-3ac1-bff0-d46f668e506a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0fa6c4a5-6122-3557-b557-39b80b3896c5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c63bbe2f-cade-33a9-9822-18d1bbf415e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_baf2a033-4368-38d1-80a6-89b65e10a63f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bf564bc1-d623-32dd-836c-6492874f9cfa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_deef28d5-b406-3997-b4fe-6c562ec948d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_27ff494d-11f7-3d29-8181-82b127bf89f6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_543acae1-ca45-3d70-83da-88e67e2cad95_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cf4ead5e-28e4-3f62-8aea-510847eaacaf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b3a7cf01-e4ac-346f-9268-e2553e2ee883_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eeb7561e-d2db-39c0-b555-8992890c484b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_711013f3-561c-324b-a8b9-669478719708_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c0734f8d-1190-3003-8d7d-4efcaa0b05d1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d5c1191c-6b12-32b3-b3a8-c6a2f9bd61cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ddf57590-cfae-345e-89b1-963231f36786_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f2a1945f-e207-3f54-b76c-4641b2dded98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_47b8aa7c-53f2-3c66-9e2d-443f841d3145_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0cc07746-3a01-3515-8cfb-409e41da3948_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eaab0259-a5ea-3009-8a32-49307ab22ccf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1e3e2fd9-2457-34f8-960f-10641a82c4ab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_919bcdc6-592c-367b-b5a9-97387e26d924_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b79f9a12-e3a3-3408-888a-eb6d78a4347c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_25ee9142-c712-3c7e-b250-05992b98afcd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3b36f4c-219a-377b-b2b0-07a04655d5df_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3b511d65-82cf-3319-a253-c70200b9d384_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fb362cc1-9672-3029-8c57-54782d3470b5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_236d1415-0304-309e-9cd7-e25c1d3841b2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3c49d79-2d88-3856-a575-c47453d28962_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16aef35b-523a-3113-91cc-334a99e6b85a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f506c8a1-5f0c-35b1-a65e-6254affc554b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a8606a25-40c7-397a-9450-28494c81958d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7ff61cbd-549b-3bf9-87f9-84171a27e638_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b701fc10-d389-319e-b3c5-f32784fa64ab_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ef2f0c0e-a6fe-3a06-b768-d775f0218580_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_42801399-7b0d-3ca2-b4ef-b95cf2f3b9d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8c9661d8-4bf7-322c-a306-8890dec3fcfb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7a48189-8b2e-33c3-ba02-8ac459fb3b4e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_60a6cd82-105e-3bb3-b571-886341927858_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_158f9b94-1f73-3229-b38e-a07be65b0da6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_345fa2d3-1a3a-3a92-8bf7-db97f5cb7a32_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_37b9d57e-92f8-30fb-a66d-852b4eecc0af_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ffba169-835b-308b-8911-66a94602c990_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bc0f3b5a-7893-3086-afb6-a31bca58f77e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_05c63987-3a53-3ba9-a274-2ddddab9257a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d806414c-987e-32f8-8d01-c35e0fab9bf1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45745f17-cca4-3617-ab80-4f0299d277b5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_27b7733a-3e36-3fdb-b1e5-1deb8d588380_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b3dd761c-729e-354f-9aa1-023c2a6e63c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_227d4d96-995a-35b3-a486-74b442984fdb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ea78a762-28c4-38e0-8205-f4fc33b9c44a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b83e4b6c-742c-3c45-8dc8-a6b42d05cb34_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_13bb33d2-9d85-37f6-b94f-6317cb9b679a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_743eb699-d26e-3222-92be-fd3f723ab740_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7193a844-9608-3216-a31f-bf508b4fe3c1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d5e6a48f-026b-3992-85a1-857599370219_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00919e4f-e8f2-314b-8279-c93868a09025_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4dbee59a-154c-3416-9cee-c327aaaaa3cb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a0dc47bd-4f93-319c-8a08-038a5bcb7f27_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_691571ed-aeaa-3749-8ee2-fd467d8358ef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8bc57672-f17b-32cd-934d-9ccb7d4307ae_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c8b02ce8-12e5-3070-a55d-2c03360ab268_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7eca3686-91fe-33c8-a58e-0b01a558854d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9b0e843-58ec-35b0-a0e0-4cc1f6469ab0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a319894-0bbc-34fb-8db9-efa016ff21e5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_04c86dd7-90f2-35ab-b040-c35b592951ff_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ce999f1-0ec3-3c1a-8598-3fb84c8719cb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fb66e958-a847-3fad-86d9-f507ddb4fe43_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2a9b90a6-0b75-34a8-8126-d9e1c37fd4b5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7d5176d7-6da3-313d-8769-d3aecf1fe786_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_de1483d1-40b8-3fc4-9c84-d80e1c4327fa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7896de19-af7e-3fe5-81b3-039ea0cf7f16_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b4d63397-bb3d-3748-b204-30ec98411649_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3456a60-d638-3674-8035-6d7568ca0dc8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c9fd5166-8297-3c55-af9a-186a885a620b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2507bf19-68f7-3c8d-a27a-97f6ff76a3b1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f682515b-4334-3531-9414-bb959268e120_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e4939d20-8040-38f5-a6f5-8921b5ef1d32_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7610e6bd-8b91-34ef-92f3-724eb3d82cbe_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d57587f-cd19-3331-8870-4df3f234e871_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_775456d9-d5d8-351b-a211-bbc8ad0d65a3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6fd097e1-ee9a-3c55-94fe-10106b09d7b2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cfdbb0a7-e6e9-3e30-80f4-781c5737e686_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db7b806d-63b3-3421-bca1-2a3e7ec14586_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6922642f-1219-31f6-9a84-3edd6e51979d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d6a2088-78d5-34c4-9350-5150c90da37d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ad32d290-5799-3d44-a9a0-6a0ece8ca26b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_123a8f54-bc47-3165-8f8c-69c73ccee912_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5616c2d4-8a50-36a6-aa01-216f77406633_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_131b21d4-db5a-3ef8-8da5-b7e529556115_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_32e313ac-cc50-3d65-99dd-0a0093328c33_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_77499fd7-62ab-3ab8-a1d5-7577624c88a1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_49a69134-d399-3cdd-8b3c-d0cab6ae13ac_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_39293742-176e-3843-9122-6da7f47aab2b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1ed1483-1183-35a8-8161-bf5e46f09824_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae5a7166-05f0-3b61-8e9e-aaa6280713f3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_103ef125-fdbd-31ba-a4e9-64d51a66c7a1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d50a221a-c3fc-3442-920b-9c056839b1e5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_caf8e46f-fe85-32ea-a422-cacdc6140eb5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_46969014-2e38-32c6-8046-abfb88d9ca71_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0b922255-8610-3321-b07b-9c22e72a1b51_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4c070bcf-57f8-375a-8c62-d629c2c6a4af_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3ab9843d-abd3-3e4f-9219-e7cac7ba85d6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_866106a4-ba10-369d-8dc5-78f0a541cd89_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_84845cbb-95c9-3bbc-9b0d-5f89d0fd0b9d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd8d8cb4-4ab7-3387-a20f-3fb35202c3f6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_037bae30-53b2-3e6b-93c9-3b339b01f76d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_923b1c23-8d5c-3ab7-b496-af952d973faf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c8820ea5-4fd4-3a42-878e-f397aaa606c4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_14244471-8892-3b79-a38f-0654a4b19c36_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3dfec499-a4eb-37ad-b832-79e2f535e527_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c24a8e5c-fc92-3964-ae06-c98df0e71a51_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f501c4f8-f852-3253-a2b2-ef8a38aff907_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d02b01a8-fa92-3111-9fd6-1077c443480c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f19148f3-ca39-3e7a-9104-fe57ec197eef_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e45d995d-90cc-3bed-85ee-89ba2cd62a59_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4da63258-18aa-32fb-a4a1-2e6333e15c06_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0608f775-5d21-3551-86aa-d269bba4e0fd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a50bd40-39b8-3c21-941f-c445cc116b18_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_71ec2c94-8ebf-3974-a20b-efd7fcffe3af_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_94d11613-1f30-364a-8e63-a20226ecbfd7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_207983a3-6f6b-3d09-9f13-14e9f3386931_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ae953c6d-ec64-3001-9307-c62de59e15e7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4493d27d-8fe0-315f-ad91-c8408a6a821e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bec1b5c8-b8b3-3d2a-bd3c-1a7b526ff488_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_50e879f7-2eb7-3a05-b0ab-8b4a5d87e132_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_76c1a411-6dee-3c3e-9c2b-2d2aff1762af_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_293ea443-6486-3d76-bb8f-f1e630a4c093_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1b23992-fb6c-32af-8373-ef387a625710_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_586422d8-80e3-30b1-a572-714b64f506fa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_caf9f790-ee04-32cc-80ad-36782a58097a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_72d861c8-10fb-3766-bdaf-ae6029c90ad3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7147c5d7-4764-32b7-a8a0-bc8f21ca852e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1f7971f0-9fb1-39ad-981e-eca0a87cdbf3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b4a170cf-dc4c-3dfb-8d45-895b69f0df83_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_79e15c4d-d4af-3c1a-ba03-7ff3fd6556e1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2ebf1097-fd86-3575-8578-524fc2990e25_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6638b296-78e1-3a83-a6c3-2c432d7b2085_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_290ea16e-1e36-3a4e-a8d3-dc105f8a66d1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cf27e7a6-c70c-3c2a-8b02-f89518f347b6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_847b007b-cf59-3ec8-8645-1334ff0ba478_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_64ee53b1-b510-3d41-b4f8-3272b26c6a8d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_adc9cc24-84cd-3768-9656-7ded27ea8b22_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b6b212c2-0e1f-3a81-be64-e40302999b40_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_132930f2-6036-30d9-a3b4-b37540c362ca_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7610f352-36e1-3561-bd61-8ca688883104_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_57f737a8-42f8-3c48-aaa3-8577a96833db_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16877f3e-7619-378f-8c79-986f9b652ef6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cae0a8da-d83f-3f5b-a4c3-3fad7bf16060_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7169d829-d2f6-3187-96f1-30688793f4cf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_902deec1-4e2a-35e2-9ee6-33ba20998878_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f21f10a8-1187-3774-b065-ff1922d40105_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_759c0da5-302b-3036-8891-c7cec954b88e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1acab93c-5709-38c9-ab19-f9be775ed184_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_69a29a97-f533-3097-9dae-78998909d660_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_90620a19-4266-337f-94f5-5ff7dec70b9e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_df4b0835-48f4-330e-a598-64e776ebfee8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b979a642-1442-320d-a3ab-7b5ef44faf3b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d981bc78-05e3-341c-b394-80a7d27c331e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c19e170b-2efd-3c9f-933c-0e9988c55f87_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8d69f4f5-13e5-3053-9789-782243557e6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9e4dadc6-3348-3b8a-8d3a-35ec4adc3196_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_da2a03a9-eb1c-3d9a-9d33-fa0e108f04a0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_589f9979-e491-3028-8cab-644ade39e456_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8b0ef9c5-ce32-3160-8332-5877625b79e2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_60039673-930a-326a-84d9-8a9ac7374aaf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c6f38d78-4c47-30d3-9eeb-28d6f357a4fc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c4435b29-37eb-3dff-aac9-3ea8a4b05faf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45beeab5-708a-32ef-8579-17867a2147d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a50ad58-dd57-3351-8020-13469f6ec5d5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_159610e5-ad9a-3b4f-89db-7e8cf3c3e4ad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07a5600b-7fb6-36e6-aa29-1ccdc3846387_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_76e17fbf-56c7-3f47-b42d-fae724941c07_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_582bd575-92e8-3887-8f99-d24920afcb3c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ca89aa40-4a4f-3c79-98a2-fc3ea39edce6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4557bbd6-9993-3d91-af11-8aa607dd966c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac84f6f9-8ea2-39db-aff8-c091a7314017_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0587630d-331c-34ed-a486-fd82b0cd09f2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f4f34f78-0255-3858-88d3-07a668a1fec6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d5c938d-029c-3520-b98f-7d86803e389a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_afe75a74-0f3e-3918-ab3c-dad55036840f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_20949467-3ec2-3844-b193-7cdbbae332c6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8f2f6ab-33ff-3d6e-85a9-182fbf8d7fa6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6a3d58a4-e9bf-3329-97d6-57c0ed437ad6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ad6a154-c436-35cc-b2f5-46aae7442bc2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_07667641-56f6-3c04-b9e4-99e58c6a1bb3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8840a0c3-ead2-3fb9-8eea-9fb0fb4c78bb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e053d002-ad19-369e-954b-aaf9e083049e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9ec5d48-a429-3754-8401-b4aee6cf3df0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45a4b513-5587-378c-87b8-2e4f1c5bf988_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f51dd019-da22-3395-b7ce-260b69d2a259_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_150159b7-96d0-3495-a6f9-b65f84940270_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_45bc7acc-1ac1-3263-aaff-ef8d50fdeee3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9a26782f-5911-3a01-87a7-3c5dfaf94056_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b5895c9f-03a2-399a-8c29-a81697c1199a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61d735ac-7e18-3d1c-989d-f4f7f5b8712c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_254b5afd-3111-39e5-b502-57aa53acf581_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_86fdd40b-b3b9-3514-9a6c-3b289132d760_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1ea26b84-7c0b-3c23-a76b-3119799236ed_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_15500ecc-93d2-3439-89c7-56a7eee845c3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3aaa9bd7-5686-3b1c-b43b-9a50c598bbf7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a46ddce6-64ca-3775-9df4-fc5e5dec5d93_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8ac9fe80-1390-364e-a31c-4bbc91cdd1fb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_295ba321-8676-3744-897e-a460ce2e9b70_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c72654cd-2d58-3b2c-8f8f-7dd23b0a54e3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_76c24827-b781-32df-9997-af9aeac2cbd7_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7a98b69b-75ad-38a9-87c0-856b08f062ec_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_53a076ec-915d-3cbd-8afe-bbe32048c468_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9ef0f936-b5d5-34f8-bf3f-a8b3197346ea_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c3dace06-001e-341b-90b9-6552715767b2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_16429cbe-b6c3-3249-8ff2-284c93be9698_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4a75d936-c926-3449-82ad-efea3b8be4f8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ed8f6314-69b4-3d9f-9834-8eb0f328fdee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_ac170d06-f67a-3234-9ffc-8b8e7ec9d0ce_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f433a129-3f9e-3454-a516-714dda5b3937_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f86f80cf-1f3a-3a7b-9e64-91fba785b3d5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a05b00bf-4ab0-3ca0-a411-7f5b6f75ee1a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_18dfb3af-fe79-365e-acbd-c11c891af004_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_78a79a18-e07b-39ea-a04b-0435b16235f1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3828cd5d-f2e0-364f-960b-e12eb1b80f17_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_55c541ec-7773-3eff-9d94-60e0ac14057c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fdf24ca4-d3be-3000-a270-391ddeb6d1e4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_29b4bdea-ecae-375c-b2f1-f9830a0c85bd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_23682182-2544-3165-8d04-c8f7d4f4b54c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_9d38b88f-e99f-3821-affc-4f4557691f4e_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6e58a518-4a7a-3235-98be-1f3dd4cc7123_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7404c1b5-0f91-3871-adbc-ffadc7a80165_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_a4769598-08db-3b73-8049-f2458e563fd2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_08579d15-4387-344b-bd13-d6f7a2a6a4b9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8e78667c-019c-3dbd-aa2e-bc6b71b3c84b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_61c6208b-67ce-3412-98c7-fa19cf03db00_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7072ce60-a9ea-33b8-b65e-7dae0360aff2_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e3966917-a353-3bb7-914c-c18f18630b85_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b504a1d7-ae7e-36ed-908b-8441a01f48ad_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_64be3d81-10df-3618-9f46-cd567d7c0182_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8261c91a-068a-31ee-951a-1245a79e58bd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b9c7d557-246c-31bc-86a8-eaab9855a1c6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_95490191-34ad-3b95-a6c6-f05b39746a83_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_b032c866-25cc-39e1-af0c-ccb435a67c7f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e47f1a98-3f52-35dc-92e1-8eec9d40e569_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f690dc6d-1c8c-3d11-b66e-57e0329305e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d692aeab-840a-3ab1-a014-9bb502d17ba5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1a1a3f70-0d64-3665-9718-669e8de4dea5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1058cc64-9a1e-3923-8363-ee184aa5adcd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d91c56ff-cd7d-3090-a2d5-0fdfc14d530d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_db2ab241-e7e3-3b75-bb93-9d4d88bae1f3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_7195c306-5db4-336f-a715-a30d4b8b34e9_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8e639671-6aef-3997-b486-688276ad21dc_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f9e1099e-2814-324a-8f22-7a45f56f2b5c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_871b8bad-b18d-3278-b344-52e1ea9111f1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e8740b37-c72b-3698-abd5-8a93331e589d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e58cc8b1-4893-3616-8eba-818e97553273_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_885a3392-6f47-3115-a7b7-c1eb789072bf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_62cf3f79-c4d8-3dc4-8324-b7dabe10fa19_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c03cfc48-9aa5-3086-acb1-36cac52e5e03_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_bee689ae-c781-3d18-80b6-be4d8edb3222_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_507f4bd5-a009-305b-af2f-3dfc0e5ae02b_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f5b941d8-61e6-362b-abf2-1153106917be_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3c7ae04d-a1ac-360e-8945-480f7c18a932_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_0c96cf1c-e454-3bd6-9649-22c0983f26be_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2ae0daf8-80a5-354c-9524-8f1f3ffc1c74_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c1ef1fc8-6d1d-3b36-9f0e-be267ce46a71_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cdeaf001-ed8e-3641-b0ab-d2dc50029d08_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_00b0f4e8-5a5e-3183-a8fc-ab68c2ec6438_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f21d9925-d137-3570-a5f0-3b521875af3c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c94d0df2-7169-320f-8bad-ba4461e46b62_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fdf32da3-ce26-3615-937f-878d21996bd8_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cecf61b8-977e-3cfb-852a-fdc64549563a_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c2f1637c-6662-3de5-a13c-368942240256_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_d7d44661-b924-3ca3-8df1-e2e5b91a6d51_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_aecb819d-7cda-39b0-88ae-44d64e88db52_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_44fbc4e4-a8f6-3825-9894-3de0ef907731_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_c3c2c07d-a505-3e95-9778-0b3b8ac4d24c_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_8f7a235c-1133-3a1b-95f1-6dbda377f840_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_909c232f-a3d5-3eb2-a5ff-0ae3818d38d6_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_dd6f8858-3040-322a-8d99-95ee28d353c1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_df3e64f2-1257-38aa-9ec4-5c6c7e107ddb_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_eec58aa7-04ba-3718-997a-7bf68a07aff3_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_1d448f3d-def9-360d-8fd9-1059566089fd_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_2dfacad0-63c9-319c-bc15-fe573d1f12d0_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4d3f0da4-d59a-3019-b3f8-bf8ea739ed6f_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_38daa741-e6ad-3e65-b70d-4a6ec19237aa_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_f8595639-ea73-301a-9c1c-97cd40d409d4_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fb50657a-44f5-3c99-a513-2712bbc469c5_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_4389656b-2857-3809-9874-f7522e5e52bf_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_e58c3df0-6af6-36f3-9274-59d14ae05169_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_78a26116-f4e0-3d07-9eba-9f8f0e17c433_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_018dbe9b-4232-3d3a-9966-0be269e38c98_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_3fa8dc41-9019-321a-aeb9-232d5aa11c42_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_922e6b32-ac7c-3b3a-9e5f-a3e0f06be6ac_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_fe09eba3-e170-303e-a96d-3856599565ee_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_923532d4-7216-3c1d-818f-9339747bdb7d_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_94895b00-4260-37bd-b61e-318a0233c145_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_296c7842-0945-3726-b753-d8bccc47bc49_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_5df50ffa-f2b9-3e14-9382-7d3e0705aa23_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_cccf5ef1-ce1b-3d5a-95c5-a28c5b1417ea_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_6534bd62-cbe8-3e3c-9974-9235fe1b4db1_R' while harvesting metadata. imsmanifest.xml: Failed to locate resource 'I_97d210d4-86f0-365a-87d3-2df8e6668be2_R' while harvesting metadata. ----- VALID! ----- Schema Location Results Schema locations are valid. ----- VALID! ----- Schema Validation Results The document is valid. ----- VALID! ----- Schematron Validation Results The document is valid. Curriculum Standards: Analyze functions that include absolute value expressions. - HSM.A1.5.1 Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. - G.GGMD.4 Graph and apply piecewise-defined functions. - HSM.A1.5.2 Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. - G.GGMD.2 Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. - G.GGMD.3 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. - CAG.M.GHS.30 Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. - G.GGMD.1 Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. - A1.A.4.3 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. - LER.M.A1HS.12 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - LER.M.A1HS.13 Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. - A1.A.4.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.) Instructional Note: Address this standard before discussing exponential functions with continuous domains. - LER.M.A1HS.11 Graph and apply step functions. - HSM.A1.5.3 Graph and analyze transformations of the absolute value function. - HSM.A1.5.4 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - MAFS.912.F-IF.3.9 Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). - CAG.M.GHS.29 Add, subtract, and multiply polynomials. - HSM.A2.3.2 Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - MAFS.K12.MP.3.1.a Prove and use polynomial identities. - HSM.A2.3.3 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - MAFS.912.F-BF.1.2 Predict the behavior of polynomial functions. - HSM.A2.3.1 Model and solve problems using the zeros of a polynomial function. - HSM.A2.3.5 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f-_, and the range of the function f is the domain of the inverse function f-_. - A2.F.2.3 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. - ETD.M.GHS.25 Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. - A2.F.2.4 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. - ETD.M.GHS.26 Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. - A2.F.2.1 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - ETD.M.GHS.27 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. - ETD.M.GHS.28 vertical and horizontal asymptotes; - F.AII.7.i end behavior; - F.AII.7.h Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. - MAFS.912.G-SRT.1.2 determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; - EI.A.6.a composition of functions algebraically and graphically. - F.AII.7.k Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - MAFS.912.G-SRT.1.3 write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and - EI.A.6.b intercepts; - F.AII.7.e graph linear equations in two variables. - EI.A.6.c zeros; - F.AII.7.d connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; - F.AII.7.g values of a function for elements in its domain; - F.AII.7.f domain, range, and continuity; - F.AII.7.a Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - CAG.M.GHS.31 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. - CAG.M.GHS.32 extrema; - F.AII.7.c Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. - CAG.M.GHS.33 intervals in which a function is increasing or decreasing; - F.AII.7.b Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1 Solve an equation of the form ??(????) = ?????? for a simple function f?????? that has an inverse and write an expression for the inverse. Example: For example, ??????????(????????????) =2 ??????????????³ or ????????????????(??????????????????) = (????????????????????+1)/(??????????????????????–1) for ???????????????????????? ? 1. - MAFS.912.F-BF.2.4.a Solve systems of linear equations using the substitution method. - HSM.A1.4.2 Solve systems of linear equations using the elimination method. - HSM.A1.4.3 Describe the effect of the transformations ??????????????????????????????????????????????????????(??????????????????????????????), ????????????????????????????????(??????????????????????????????????)+????????????????????????????????????, ??????????????????????????????????????(????????????????????????????????????????+??????????????????????????????????????????), and combinations of such transformations on the graph of ????????????????????????????????????????????=??????????????????????????????????????????????(????????????????????????????????????????????????) for any real number ??????????????????????????????????????????????????. Find the value of ???????????????????????????????????????????????????? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) - A1.FBF.3 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??????????????????????????????????????????????????????(????????????????????????????????????????????????????????) gives the number of person-hours it takes to assemble ?????????????????????????????????????????????????????????? engines in a factory, then the positive integers would be an appropriate domain for the function. - MAFS.912.F-IF.2.5 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. - MAFS.912.F-IF.2.6 Graph solutions to linear inequalities in two variables. - HSM.A1.4.4 Graph and solve a system of linear inequalities. - HSM.A1.4.5 Find the zeros of quadratic functions. - HSM.A2.2.3 Solve problems with complex numbers. - HSM.A2.2.4 Identify key features of quadratic functions. - HSM.A2.2.1 Write and graph quadratic functions in standard form. - HSM.A2.2.2 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - MAFS.912.F-IF.2.4 Solve linear-quadratic systems. - HSM.A2.2.7 Solve quadratic equations by completing the square. - HSM.A2.2.5 Solve quadratic equations using the Quadratic Formula. - HSM.A2.2.6 Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. - A2.F.1.4 Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. - A2.F.1.2 Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. - A2.F.1.3 Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. - A2.ASE.1 Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. - A2.F.1.8 Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2 solve multistep linear inequalities in one variable algebraically and represent the solution graphically; - EI.A.5.a solve practical problems involving inequalities; and - EI.A.5.c Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) - A1.ACE.1 Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). - PRR.M.A2HS.7 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) - A1.ACE.2 Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. - A1.ACE.4 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. - PRR.M.A2HS.9 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. - MAFS.912.G-SRT.2.4 Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). - PRR.M.A2HS.4 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - MAFS.912.G-SRT.2.5 intercepts; - F.A.7.d values of a function for elements in its domain; and - F.A.7.e connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. - F.A.7.f Factor a quadratic trinomial. - HSM.A1.7.5 Factor a quadratic trinomial when a ? 1. - HSM.A1.7.6 determining whether a relation is a function; - F.A.7.a domain and range; - F.A.7.b Factor special trinomials. - HSM.A1.7.7 Combine like terms to simplify polynomials. - HSM.A1.7.1 Multiply two polynomials. - HSM.A1.7.2 Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. - A1.A.2.2 Use patterns to multiply binomials. - HSM.A1.7.3 Factor a polynomial. - HSM.A1.7.4 Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. - 9.2.1.6 evaluate algebraic expressions for given replacement values of the variables. - EO.A.1.b Make qualitative statements about the rate of change of a function, based on its graph or table of values. - 9.2.1.8 Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations - 9.2.1.9 The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. - EO.AII.2 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ???????????????????????????????????????????????????????????? is a function and ?????????????????????????????????????????????????????????????? is an element of its domain, then ????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????) denotes the output of ???????????????????????????????????????????????????????????????????? corresponding to the input ??????????????????????????????????????????????????????????????????????. The graph of ???????????????????????????????????????????????????????????????????????? is the graph of the equation ?????????????????????????????????????????????????????????????????????????? = ????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????). - MAFS.912.F-IF.1.1 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - MAFS.912.F-IF.1.2 Relate roots and rational exponents and use them to simplify expressions and solve equations. - HSM.A2.5.1 represent verbal quantitative situations algebraically; and - EO.A.1.a Perform operations on functions to answer real-world questions. - HSM.A2.5.5 Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. - A1.SPID.6 Graph and transform radical functions. - HSM.A2.5.3 Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. - A1.SPID.7 Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain - 9.2.1.1 Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.) - A1.FLQE.2 Distinguish between functions and other relations defined symbolically, graphically or in tabular form. - 9.2.1.2 Find the domain of a function defined symbolically, graphically or in a real-world context. - 9.2.1.3 Represent the inverse of a relation using tables, graphs, and equations. - HSM.A2.5.6 Obtain information and draw conclusions from graphs of functions and other relations. - 9.2.1.4 Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - A1.FLQE.5 Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form - 9.2.1.5 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - CCSS.Math.Content.HSG-GMD.A.3 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. - AP.M.GHS.48 The student will solve problems involving equations of circles. - PC.G.12 Explain and use the relationship between the sine and cosine of complementary angles. - MAFS.912.G-SRT.3.7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - MAFS.912.G-SRT.3.8 Use permutations and combinations to compute probabilities of compound events and solve problems. - AP.M.GHS.50 Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. - A2.AAPR.1 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. - MAFS.912.G-SRT.3.6 Use properties of exponents to solve equations with rational exponents. - HSM.A1.6.1 Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. - LER.M.A1HS.32 Describe and graph exponential functions. - HSM.A1.6.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship or two input-output pairs (include reading these from a table). Instructional Note: In constructing linear functions, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions. - LER.M.A1HS.30 Use exponential functions to model situations and make predictions. - HSM.A1.6.3 Solve equations involving several variables for one variable in terms of the others. - A1.A.3.1 Identify and describe geometric sequences. - HSM.A1.6.4 Perform, analyze, and use transformations of exponential functions. - HSM.A1.6.5 Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. - A1.AREI.1 applying the laws of exponents to perform operations on expressions; - EO.A.2.a Determine the maximum or minimum value of a quadratic function by completing the square. - A2.ASE.3b add, subtract, multiply, divide, and simplify rational algebraic expressions; - EO.AII.1.a add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and - EO.AII.1.b Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Instructional Note: Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. - LER.M.A1HS.27 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. - LER.M.A1HS.28 factor polynomials completely in one or two variables. - EO.AII.1.c Recognize that geometric sequences are exponential using equations, tables, graphs and verbal descriptions. Given the formula f(x) = a(r)x, find the next term and define the meaning of a and r within the context of the problem. - A1.A.3.6 Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1 Solve systems of linear equations using the substitution method. - A1.AREI.6 Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. - A1.AREI.5 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A1.AREI.3 identifying the converse, inverse, and contrapositive of a conditional statement; - RLT.G.1.a determining the validity of a logical argument. - RLT.G.1.c translating a short verbal argument into symbolic form; and - RLT.G.1.b Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on linear functions and exponential functions whose domain is a subset of the integers. The Unit on Quadratic Functions and Modeling in this course and the Algebra II course address other types of functions. - LER.M.A1HS.23 Use coordinates to prove simple geometric theorems algebraically. - G.GGPE.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. - LER.M.A1HS.21 Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. - G.GGPE.5 Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. - G.GGPE.6 Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. - G.GGPE.7 Recognize that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.18 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.19 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. - LER.M.A1HS.14 use knowledge of transformations to convert between equations and the corresponding graphs of functions. - F.AII.6.b recognize the general shape of function families; and - F.AII.6.a Find the probability of an event given that another event has occurred. - HSM.G.12.2 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - MAFS.912.G-CO.1.3 Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ????????????????????????????????????????????????????????????????????????????????+?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? for real numbers ?????????????????????????????????????????????????????????????????????????????????????? and ????????????????????????????????????????????????????????????????????????????????????????. - A2.AREI.4b Rewrite and use literal equations to solve problems. - HSM.A1.1.4 Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.G.12.3 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. - MAFS.912.G-CO.1.1 Define probability distributions to represent experiments and solve problems. - HSM.G.12.4 Solve and graph inequalities. - HSM.A1.1.5 Calculate, interpret, and apply expected value. - HSM.G.12.5 Write and solve compound inequalities. - HSM.A1.1.6 Reason about operations with real numbers. - HSM.A1.1.1 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. - MAFS.912.G-CO.1.5 Create and solve linear equations with one variable. - HSM.A1.1.2 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - MAFS.912.A-SSE.2.3.b Factor a quadratic expression to reveal the zeros of the function it defines. - MAFS.912.A-SSE.2.3.a Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - MAFS.912.A-CED.1.2 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?????????????????????????????????????????????????????????????????????????????????????????? = ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? to highlight resistance ????????????????????????????????????????????????????????????????????????????????????????????????. - MAFS.912.A-CED.1.4 Write and solve absolute-value equations and inequalities - HSM.A1.1.7 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. - MAFS.912.A-CED.1.1 Interpret parts of an expression, such as terms, factors, and coefficients. - RQ.M.A1HS.4.a Use relationships among events to find probabilities. - HSM.G.12.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. - RQ.M.A1HS.5 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. - RQ.M.A1HS.6 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. - RQ.M.A1HS.8 Solve common and natural logarithmic equations using the properties of logarithms. - A2.A.1.6 Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. - A2.A.1.1 Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. - A2.A.1.2 Use dilation and rigid motion to establish triangle similarity theorems. - HSM.G.7.3 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. - EE.M.A1HS.47 Determine whether figures are similar. - HSM.G.7.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.46 Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. - HSM.G.7.5 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.45 Use similarity and the geometric mean to solve problems involving right triangles. - HSM.G.7.4 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. - MAFS.912.G-CO.2.8 Dilate figures and identify characteristics of dilations. - HSM.G.7.1 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². - EE.M.A1HS.49 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. - MAFS.912.G-CO.2.7 Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). - A2.A.1.8 Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. - A2.A.1.9 Use the volumes of right and oblique pyramids and cones to solve problems. - HSM.G.11.3 Calculate the volume of a sphere and solve problems involving the volumes of spheres. - HSM.G.11.4 Solve absolute value equations and interpret the solutions in the original context. - A1.A.1.2 Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. - A1.A.1.3 Identify space figures and their relationships with polygons to solve problems. - HSM.G.11.1 Use the properties of prisms and cylinders to calculate their volumes. - HSM.G.11.2 Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. - A1.A.1.1 Add, subtract, multiply, divide, and simplify polynomial and rational expressions. - A2.A.2.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - A2.A.2.4 The student will use surface area and volume of three-dimensional objects to solve practical problems. - TDF.G.13 Interpret parts of an expression, such as terms, factors, and coefficients. - PRR.M.A2HS.6.a Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. - A2.A.2.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. - MAFS.912.A-APR.1.1 Use trigonometry to solve problems. - HSM.G.8.5 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. - CPC.M.GHS.9 Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - HSF-LE.B.5 Use trigonometric ratios to find lengths and angle measures of right triangles. - HSM.G.8.2 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.7 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.8 Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. - HSM.G.8.1 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle) - CPC.M.GHS.5 Use the Law of Cosines to solve problems. - HSM.G.8.4 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.6 Use the Law of Sines to solve problems. - HSM.G.8.3 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - CCSS.Math.Content.HSG-SRT.B.5 Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers. - G.SPID.2 A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. - MAFS.912.G-SRT.1.1.a Determine whether a function is a relation. - HSM.A1.3.1 Identify, evaluate, and graph linear functions. - HSM.A1.3.2 Transform linear equations - HSM.A1.3.3 Identify and describe arithmetic sequences. - HSM.A1.3.4 Interpret the parameters in a linear or exponential function in terms of a context. - MAFS.912.F-LE.2.5 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle). - CPC.M.GHS.3 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. - CPC.M.GHS.1 The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. - S.AII.9 The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - MAFS.912.G-SRT.1.1.b Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) - A1.ASE.1 Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. - HSM.A2.1.1 Use graphs and tables to approximate solutions to algebraic equations and inequalities. - HSM.A2.1.5 Apply transformations to graph functions and write equations. - HSM.A2.1.2 Graph and interpret piecewise-defined functions. - HSM.A2.1.3 Use a variety of tools to solve systems of linear equations and inequalities. - HSM.A2.1.6 Solve systems of equations using matrices. - HSM.A2.1.7 The student will verify and use properties of quadrilaterals to solve problems, including practical problems. - PC.G.9 Interpret the parameters in a linear or exponential function in terms of the context. - A2.FLQE.5 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. - CCSS.Math.Content.HSG-SRT.A.2 Use geometric shapes, their measures, and their properties to describe real-world objects. - G.GM.1 Prove that all circles are similar. - CWC.M.GHS.34 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - CWC.M.GHS.36 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. - CWC.M.GHS.35 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - MAFS.912.A-REI.2.3 Describe the effect of the transformations ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????), ????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????)+????????????????????????????????????????????????????????????????????????????????????????????????????????????, ??????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????+??????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and combinations of such transformations on the graph of ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) for any real number ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????. Find the value of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3 Use the equations and graphs of parabolas to solve problems. - HSM.G.9.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? - CCSS.Math.Content.HSS-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. - CCSS.Math.Content.HSS-MD.A.3 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. - CCSS.Math.Content.HSS-MD.A.2 Use the coordinate plane to analyze geometric figures. - HSM.G.9.1 Use the equations and graphs of circles to solve problems. - HSM.G.9.3 Prove geometric theorems using algebra and the coordinate plane. - HSM.G.9.2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. - CWC.M.GHS.41 Write and graph linear equations using point-slope form. - HSM.A1.2.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). - MAFS.912.F-LE.1.2 Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. - CWC.M.GHS.40 Write and graph linear equations using standard form. - HSM.A1.2.3 Write equations of parallel lines and perpendicular lines. - HSM.A1.2.4 For exponential models, express as a logarithm the solution to ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? to the ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? power = ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? where ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, and ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? are numbers and the base ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? is 2, 10, or ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????; evaluate the logarithm using technology. - MAFS.912.F-LE.1.4 Write and graph linear equations using slope-intercept form. - HSM.A1.2.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. - MAFS.912.N-RN.1.1 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - MAFS.912.N-RN.1.2 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) - A2.AREI.7 Identify the effect on the graph of replacing ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) by ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) + ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? + ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) for specific values of ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? (both positive and negative); find the value of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. - MAFS.912.F-BF.2.3 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. - CCSS.Math.Content.HSG-CO.C.10 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. - CWC.M.GHS.38 Construct a tangent line from a point outside a given circle to the circle. - CWC.M.GHS.37 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. - CWC.M.GHS.39 the Pythagorean Theorem and its converse; - T.G.8.a properties of special right triangles; and - T.G.8.b trigonometric ratios. - T.G.8.c Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - MAFS.912.A-REI.3.5 Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. - G.GCO.1 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? = –3???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and the circle ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² + ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = 3. - MAFS.912.A-REI.3.7 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - MAFS.912.A-REI.3.6 Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. - G.GCO.3 Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. - G.GCO.2 Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. - G.GCO.5 Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c - G.GCO.7 Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. - G.GCO.6 Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. - G.GCO.9 Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. - G.GCO.8 Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. - MAFS.912.G-CO.3.10 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. - MAFS.912.G-CO.3.11 Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts. - A1.F.1.2 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. - MAFS.912.N-RN.2.3 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - MAFS.912.S-ID.3.7 (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² + 4 as (???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? + 2??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)(???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? – 2??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????). - MAFS.912.N-CN.3.8 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - MAFS.912.G-GMD.2.4 Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. - MAFS.912.F-IF.3.7.e Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. - CCSS.Math.Content.HSG-CO.B.6 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. - CCSS.Math.Content.HSG-CO.B.7 Graph linear and quadratic functions and show intercepts, maxima, and minima. - MAFS.912.F-IF.3.7.a Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - CCSS.Math.Content.HSG-CO.B.8 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - MAFS.912.F-IF.3.7.b Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - MAFS.912.G-CO.4.13 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. - MAFS.912.G-CO.4.12 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. - A2.ACE.2 Write linear functions, using function notation, to model real-world and mathematical situations. - A1.F.1.3 Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) - A2.ACE.3 Interpret parts of an expression, such as terms, factors, and coefficients. - MAFS.912.A-SSE.1.1.a Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. - A2.ACE.1 (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. - MAFS.912.S-CP.2.9 (HONORS ONLY) Apply the Addition Rule, ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? or ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) = ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) + ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and interpret the answer in terms of the model. - MAFS.912.S-CP.2.7 Find the conditional probability of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? given ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? as the fraction of ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????’s outcomes that also belong to ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, and interpret the answer in terms of the model. - MAFS.912.S-CP.2.6 Use inverse functions to solve problems. - HSM.A1.10.7 Add, subtract, and multiply functions. - HSM.A1.10.6 Change functions to compress or stretch their graphs. - HSM.A1.10.5 Graph and analyze transformations of functions. - HSM.A1.10.4 Identify the function family when given an equation or graph. - HSM.A1.10.3 Identify the key features of the cube root function. - HSM.A1.10.2 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. - F.AII.8 Describe the key features of the square root function. - HSM.A1.10.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. - MAFS.912.G-GMD.1.1 (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - MAFS.912.G-GMD.1.2 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - MAFS.912.G-GMD.1.3 Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. - 9.4.3.6 Understand and use simple probability formulas involving intersections, unions and complements of events. - 9.4.3.7 Describe the properties of a figure before and after translation. - HSM.G.3.2 Apply probability concepts to real-world situations to make informed decisions. - 9.4.3.8 Identify different types of symmetry in two-dimensional figures. - HSM.G.3.5 Identify different rigid motions used to transform two-dimensional shapes. - HSM.G.3.4 Use the relationship between conditional probabilities and relative frequencies in contingency tables. - 9.4.3.9 Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. - 9.4.3.2 Draw and describe the reflection of a figure across a line of reflection. - HSM.G.3.1 Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. - 9.4.3.5 Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. - G.2D.1.2 Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. - G.2D.1.3 Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. - G.2D.1.4 Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. - G.2D.1.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). - MAFS.912.G-GPE.2.5 Apply the properties of polygons to solve real-world and mathematical problems involving perimeter and area (e.g., triangles, special quadrilaterals, regular polygons up to 12 sides, composite figures). - G.2D.1.6 Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. - G.2D.1.7 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - MAFS.912.G-GPE.2.6 Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). - G.2D.1.8 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - MAFS.912.G-GPE.2.7 Use numeric, graphic and algebraic representations of transformations in two dimensions, such as reflections, translations, dilations, and rotations about the origin by multiples of 90?, to solve problems involving figures on a coordinate plane and identify types of symmetry. - G.2D.1.9 Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. - 9.3.4.1 Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - MF.M.A2HS.30.b Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). - MAFS.912.G-GPE.2.4 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - MF.M.A2HS.30.a Organize data in two-way frequency tables and use them to make inferences and generalizations. - HSM.A1.11.5 Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. - G.2D.1.1 Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. - 9.3.4.6 Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. - 9.3.4.7 Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. - 9.3.4.4 Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. - 9.3.4.5 Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. - 9.3.4.2 Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. - 9.3.4.3 Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays. - 9.4.2.1 Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation. - 9.4.2.2 Use triangle congruence to solve problems with overlapping triangles. - HSM.G.4.6 Identify congruent right triangles. - HSM.G.4.5 Apply theorems about isosceles and equilateral triangles to solve problems. - HSM.G.4.2 Prove that all circles are similar. - MAFS.912.G-C.1.1 The student will compute and distinguish between permutations and combinations. - S.AII.12 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. - MAFS.912.G-C.1.2 Use a composition of rigid motions to show that two objects are congruent. - HSM.G.4.1 Determine congruent triangles by comparing two angles and one side. - HSM.G.4.4 Use SAS and SSS to determine whether triangles are congruent. - HSM.G.4.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - MAFS.912.G-C.1.3 (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. - MAFS.912.G-C.1.4 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - SPT.M.GHS.21 Use relationships between circles, angles, and arcs. - HSM.G.10.4 Explain and use the relationship between the sine and cosine of complementary angles. - SPT.M.GHS.20 Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. - HSM.G.10.5 Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. - G.RL.1.1 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. - SPT.M.GHS.24 Solve quadratic equations by inspection (e.g., for ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ± ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? for real numbers ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????. - MAFS.912.A-REI.2.4.b Prove the Laws of Sines and Cosines and use them to solve problems. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. - SPT.M.GHS.23 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. - SPT.M.GHS.22 Use the relation ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - MAFS.912.N-CN.1.2 Use the method of completing the square to transform any quadratic equation in ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? into an equation of the form (???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)² = ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? that has the same solutions. Derive the quadratic formula from this form. - MAFS.912.A-REI.2.4.a Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. - G.RL.1.2 Find arc length and sector area of a circle and use them to solve problems. - HSM.G.10.1 Assess the validity of a logical argument and give counterexamples to disprove a statement. - G.RL.1.3 Use properties of tangent lines to solve problems. - HSM.G.10.2 Relate the length of a chord to the central angle it subtends and the arc it intercepts. - HSM.G.10.3 Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. - A1.F.3.2 Add, subtract, and multiply functions using function notation. - A1.F.3.3 ordering the angles by degree measure, given side lengths; - T.G.5.b determining whether a triangle exists; and - T.G.5.c determining the range in which the length of the third side must lie. - T.G.5.d ordering the sides by length, given angle measures; - T.G.5.a Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. - HSM.G.5.5 Use theorems to compare the sides and angles of a triangle. - HSM.G.5.4 Use perpendicular and angle bisectors to solve problems. - HSM.G.5.1 Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. - HSM.G.5.3 Use triangle bisectors to solve problems. - HSM.G.5.2 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. - MAFS.912.G-C.2.5 Solve an equation of the form ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) graphically by identifying the ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????-coordinate(s) of the point(s) of intersection of the graphs of ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) and ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????). (Limit to linear; quadratic; exponential.) - A1.AREI.11 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. - MAFS.912.G-CO.2.6 Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. - MF.M.A2HS.23 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: While functions will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra I. (e.g., Finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line). - MF.M.A2HS.24 Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. - A1.F.2.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. - MF.M.A2HS.32 Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. - G.RT.1.2 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. - MF.M.A2HS.34 Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. - G.RT.1.3 Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. - MF.M.A2HS.33 Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. - G.RT.1.4 For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. - MF.M.A2HS.36 Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (e.g., f(x) = 2 x³ or f(x) = (x+1)/(x-1) for x ? 1.) Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Extend this standard to simple rational, simple radical, and simple exponential functions; connect this standard to M.A2HS.34. - MF.M.A2HS.35 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - SPT.M.GHS.18 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. - SPT.M.GHS.17 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - SPT.M.GHS.16 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. - SPT.M.GHS.15 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. - SPT.M.GHS.19 Use the structure of an expression to identify ways to rewrite it. Example: For example, see ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????4 – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????4 as (????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????²)² – (??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????²)², thus recognizing it as a difference of squares that can be factored as (????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????²)(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² + ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????²). - MAFS.912.A-SSE.1.2 Use properties of sides, angles, and diagonals to identify a parallelogram. - HSM.G.6.4 Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. - HSM.G.6.3 Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. - HSM.G.6.6 Use the properties of rhombuses, rectangles, and squares to solve problems. - HSM.G.6.5 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. - MAFS.912.G-CO.3.9 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. - MF.M.A2HS.27 Use triangle congruence to understand kites and trapezoids. - HSM.G.6.2 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. - MF.M.A2HS.29 Find the sums of the measures of the exterior angles and interior angles of polygons. - HSM.G.6.1 Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). - G.RT.1.1 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) Note: Emphasize the selection of a model function based on behavior of data and context. - MF.M.A2HS.28 The student, given information in the form of a figure or statement, will prove two triangles are congruent. - T.G.6 The student, given information in the form of a figure or statement, will prove two triangles are similar. - T.G.7 Use matrices to represent and solve systems of equations. - HSM.A2.10.5 Write equivalent radical expressions. - HSM.A1.9.3 (HONORS ONLY) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). - MAFS.912.G-SRT.4.11 Solve quadratic equations by taking square roots. - HSM.A1.9.4 Interpret parts of an expression, such as terms, factors, and coefficients. - EE.M.A1HS.41.a Use completing the square to solve quadratic equations. - HSM.A1.9.5 (HONORS ONLY) Prove the Laws of Sines and Cosines and use them to solve problems. - MAFS.912.G-SRT.4.10 Use the quadratic formula to solve quadratic equations. - HSM.A1.9.6 Use tables and graphs to find solutions of quadratic equations. - HSM.A1.9.1 Find the solution of a quadratic equation by factoring. - HSM.A1.9.2 square roots of whole numbers and monomial algebraic expressions; - EO.A.3.a Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. - 9.2.3.5 numerical expressions containing square or cube roots. - EO.A.3.c Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. - 9.2.3.6 cube roots of integers; and - EO.A.3.b Solve a system with linear and quadratic equations. - HSM.A1.9.7 Use the unit circle to evaluate the trigonometric functions of any angle. - HSM.A2.7.3 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Instructional Note: Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. - DS.M.A1HS.38 Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. - 9.2.3.1 Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. - 9.2.3.2 Create and use graphs of sine and cosine functions. - HSM.A2.7.4 Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. - 9.2.3.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on quadratic functions, and consider including absolute value functions. - QFM.M.A1HS.58 Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. - 9.3.1.4 Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. - 9.3.1.1 Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. - 9.3.1.2 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. - QFM.M.A1HS.53 solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. - RLT.G.2.b prove two or more lines are parallel; and - RLT.G.2.a For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. - QFM.M.A1HS.51 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). - QFM.M.A1HS.50 Rewrite expressions involving simple radicals and rational exponents in different forms. - A1.NRNS.1 Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. - A1.FIF.2 Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. - A1.NRNS.2 Extend previous knowledge of a function to apply to general behavior and features of a function. - A1.FIF.1 Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) - A1.FIF.4 Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.) - A1.FIF.6 quadratic equations over the set of complex numbers; - EI.AII.3.b Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. - A1.NRNS.3 Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????+??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????.) - A1.FIF.7 Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. - G.C.1.4 Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. - G.C.1.3 Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. - G.C.1.2 solving problems, including practical problems, about similar geometric figures. - TDF.G.14.d determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; - TDF.G.14.b Use quadratic functions to model real-world situations. - HSM.A1.8.4 Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. - G.GSRT.3 Determine whether a linear, exponential, or quadratic function best models a data set. - HSM.A1.8.5 Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. - G.GSRT.4 Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. - G.GSRT.1 comparing ratios between lengths, perimeters, areas, and volumes of similar figures; - TDF.G.14.a Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. - G.GSRT.2 Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. - G.GCO.11 Explain and use the relationship between the sine and cosine of complementary angles. - G.GSRT.7 Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. - G.GSRT.8 Identify key features of the graph of the quadratic parent function. - HSM.A1.8.1 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5 Graph quadratic functions using the vertex form. - HSM.A1.8.2 Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. - A2.FIF.9 Graph quadratic functions using standard form. - HSM.A1.8.3 Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. - G.GSRT.6 Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. - G.GCO.10 The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. - EI.AII.4 Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts. - 9.2.2.5 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. - AP.M.GHS.47 Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. - 9.2.2.6 Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. - A2.FIF.6 Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. - AP.M.GHS.44 Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. - A2.FIF.5 Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. - A2.FIF.4 Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. - HSM.A2.6.4 Recognize the key features of exponential functions. - HSM.A2.6.1 Write exponential models in different ways to solve problems. - HSM.A2.6.2 Represent and solve problems in various contexts using linear and quadratic functions. - 9.2.2.1 Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. - 9.2.2.2 Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. - 9.2.2.3 Solve exponential and logarithmic equations. - HSM.A2.6.6 Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. - 9.2.2.4 investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and - RLT.G.3.c applying slope to verify and determine whether lines are parallel or perpendicular; - RLT.G.3.b determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. - RLT.G.3.d investigating and using formulas for determining distance, midpoint, and slope; - RLT.G.3.a Prove that all circles are similar. - G.GCI.1 Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. - G.GCI.3 Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. - G.GCI.2 Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. - G.GCI.5 Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. - G.GCI.4 The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - SPT.M.GHS.14.b A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. - SPT.M.GHS.14.a number of sides of a regular polygon. - PC.G.10.c Use relationships among events to find probabilities. - HSM.A2.12.1 measure of an interior and/or exterior angle; and - PC.G.10.b Find the probability of an event given that another event has occurred. - HSM.A2.12.2 Solve systems of linear equations using linear combination. - A1.AREI.6b Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - PRR.M.A2HS.2 sum of the interior and/or exterior angles; - PC.G.10.a Solve systems of linear equations using the substitution method. - A1.AREI.6a Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.A2.12.3 Define probability distributions to represent experiments and solve problems. - HSM.A2.12.4 Calculate, interpret, and apply expected value. - HSM.A2.12.5 Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. - G.3D.1.1 Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. - LER.M.A1HS.24.b Graph linear and quadratic functions and show intercepts, maxima, and minima. - LER.M.A1HS.24.a Understand and apply the geometric properties of a parabola. - HSM.A2.9.1 Understand and apply the geometric properties of a hyperbola. - HSM.A2.9.4 Write, graph, and apply the equation of a circle. - HSM.A2.9.2 the perpendicular bisector of a line segment; - RLT.G.4.b Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. - 9.3.3.1 (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. - MAFS.912.G-GPE.1.2 Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. - 9.3.3.2 a line segment congruent to a given line segment; - RLT.G.4.a Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. - MAFS.912.G-GPE.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. - A1.FIF.1a Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. - 9.3.3.7 Represent a function using function notation and explain that ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) denotes the output of function ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? that corresponds to the input ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????. - A1.FIF.1b Know and apply properties of a circle to solve problems and logically justify results. - 9.3.3.8 Understand that the graph of a function labeled as ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? is the set of all ordered pairs (??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????,????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) that satisfy the equation ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????). - A1.FIF.1c Know and apply properties of congruent and similar figures to solve problems and logically justify results. - 9.3.3.6 Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. - 9.3.3.3 Graph linear and quadratic functions and show intercepts, maxima, and minima. - QFM.M.A1HS.54.a Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. - 9.3.3.4 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - QFM.M.A1HS.54.b multistep linear equations in one variable algebraically; - EI.A.4.a Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. - 9.4.1.1 quadratic equations in one variable algebraically; - EI.A.4.b literal equations for a specified variable; - EI.A.4.c Analyze the effects on summary statistics of changes in data sets. - 9.4.1.2 Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. - 9.4.1.3 systems of two linear equations in two variables algebraically and graphically; and - EI.A.4.d practical problems involving equations and systems of equations. - EI.A.4.e Identify and describe transformations of two-dimensional figures. - HSM.G.3 Use the relationships between sides, segments, and angles of triangles to solve problems. - HSM.G.5 Write conditionals and biconditionals and find their truth values. - HSM.G.1.5 Use inductive reasoning to make conjectures about mathematical relationships. - HSM.G.1.4 Use deductive reasoning to prove theorems. - HSM.G.1.7 Use deductive reasoning to draw conclusions. - HSM.G.1.6 Use properties of segments and angles to find their measures. - HSM.G.1.1 an angle congruent to a given angle; - RLT.G.4.f Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. - MAFS.912.F-BF.1.1.b Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. - 9.4.1.4 the bisector of a given angle, - RLT.G.4.e Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - MAFS.912.G-MG.1.1 Use the midpoint and distance formulas to solve problems. - HSM.G.1.3 an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - RLT.G.4.h Use a straightedge and compass to construct basic figures. - HSM.G.1.2 arc length; and - PC.G.11.c lengths of segments formed by intersecting chords, secants, and/or tangents; - PC.G.11.b angle measures formed by intersecting chords, secants, and/or tangents; - PC.G.11.a area of a sector. - PC.G.11.d Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines. - 9.2.4.4 Solve linear programming problems in two variables using graphical methods. - 9.2.4.5 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. - 9.2.4.7 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. - 9.2.4.8 Use the Law of Sines and the Law of Cosines. - HSM.A2.8.2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - MG.M.GHS.53 Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. - A2.FBF.1b Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. - 9.2.4.1 Verify and use trigonometric identities. - HSM.A2.8.3 Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. - 9.2.4.2 Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. - 9.3.2.2 Assess the validity of a logical argument and give counterexamples to disprove a statement. - 9.3.2.3 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. - 9.3.2.1 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. - CPC.M.GHS.10 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. - CPC.M.GHS.11 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. - CPC.M.GHS.12 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. - 9.3.2.4 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. - 9.3.2.5 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. - CPC.M.GHS.13 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. - RQ.M.A1HS.10 Use slope to solve problems about parallel and perpendicular lines. - HSM.G.2.4 Solve problems using the measures of interior and exterior angles of triangles. - HSM.G.2.3 Understand and apply the relationship of rational exponents to integer exponents and radicals to solve problems. - A2.N.1.4 Simplify, add, subtract, multiply, and divide complex numbers. - A2.N.1.2 Use angle relationships to prove that lines are parallel. - HSM.G.2.2 Determine the measures of the angles formed when parallel lines are intersected by a transversal. - HSM.G.2.1 List of all Files Validated: imsmanifest.xml I_000cfdc3-d1f4-3904-9412-9d9d3c7b483c_1_R/BasicLTI.xml I_000cfdc3-d1f4-3904-9412-9d9d3c7b483c_3_R/BasicLTI.xml I_000cfdc3-d1f4-3904-9412-9d9d3c7b483c_5_R/BasicLTI.xml I_0019d27d-e20e-3e91-90d0-3b79e99d750e_1_R/BasicLTI.xml I_001ace35-75c9-3542-b440-1f394e002334_R/BasicLTI.xml I_001d33d3-c079-355c-b64c-df41d4fb255c_1_R/BasicLTI.xml I_0029230a-6f9d-30bb-a8c1-297ff8a01d04_1_R/BasicLTI.xml I_00412bbe-6e7b-3898-b260-3e5bf9466404_R/BasicLTI.xml I_00453c84-065a-3929-8e35-318613222ba2_1_R/BasicLTI.xml I_00453c84-065a-3929-8e35-318613222ba2_3_R/BasicLTI.xml I_00455151-b5de-3dd2-84cf-514b31608ca3_R/BasicLTI.xml I_0073e0b7-423b-317f-bb5f-c744bc93a774_1_R/BasicLTI.xml I_0083d2a7-798c-322a-98b3-21fd33338172_R/BasicLTI.xml I_00919e4f-e8f2-314b-8279-c93868a09025_R/BasicLTI.xml I_00958619-618b-3783-be94-db80c1951c55_1_R/BasicLTI.xml I_009f17cc-f29e-3948-8dfe-aa5b4de67b57_R/BasicLTI.xml I_00a60446-aac3-3f9b-b7e2-ecb7222841a5_1_R/BasicLTI.xml I_00a60446-aac3-3f9b-b7e2-ecb7222841a5_3_R/BasicLTI.xml I_00a9c516-f8cc-3233-9edf-8fadf30a5aff_R/BasicLTI.xml I_00b0f4e8-5a5e-3183-a8fc-ab68c2ec6438_R/BasicLTI.xml I_00b305c0-4b93-3849-bf70-7892662a54d9_1_R/BasicLTI.xml I_00cd98d0-621a-336e-95e6-778a116f78c7_1_R/BasicLTI.xml I_00e84e34-026d-3e9e-b428-311e19f54479_R/BasicLTI.xml I_00ef55aa-ed0b-39e5-9230-e5368f51c348_1_R/BasicLTI.xml I_0113ed5f-0a42-3c34-b62d-8535c2a59d06_R/BasicLTI.xml I_013062bf-c3fd-3d79-8bf9-e5c7f7c1071f_R/BasicLTI.xml I_01327d78-4344-3207-850e-87ba12921566_R/BasicLTI.xml I_01442673-a031-3940-915e-53c5989737d7_1_R/BasicLTI.xml I_0151a895-05c2-3bbb-9103-d870946d2ce1_1_R/BasicLTI.xml I_0151a895-05c2-3bbb-9103-d870946d2ce1_3_R/BasicLTI.xml I_0151a895-05c2-3bbb-9103-d870946d2ce1_5_R/BasicLTI.xml I_0158e651-84e1-3ef9-ac4d-39898c4c83af_1_R/BasicLTI.xml I_01664ddc-e1be-3241-837e-167a5ea8e5c5_1_R/BasicLTI.xml I_01664ddc-e1be-3241-837e-167a5ea8e5c5_3_R/BasicLTI.xml I_018dbe9b-4232-3d3a-9966-0be269e38c98_R/BasicLTI.xml I_01b69dac-77b8-3e69-a1c7-1f200c8f5536_1_R/BasicLTI.xml I_01b69dac-77b8-3e69-a1c7-1f200c8f5536_3_R/BasicLTI.xml I_01b69dac-77b8-3e69-a1c7-1f200c8f5536_5_R/BasicLTI.xml I_01b69dac-77b8-3e69-a1c7-1f200c8f5536_7_R/BasicLTI.xml I_01c20071-e9aa-37c9-98a7-a5bcb4119a69_1_R/BasicLTI.xml I_01dc75b5-9167-367b-b34a-7eb60123a23d_R/BasicLTI.xml I_01e384f0-192a-3d89-8b1d-7a91f5b517e2_1_R/BasicLTI.xml I_02121fc7-7ce4-32cb-9436-e5fc74feae3e_1_R/BasicLTI.xml I_02206bbd-6f62-39f1-b3de-4c217e017bdc_R/BasicLTI.xml I_02252cf0-e60c-3035-8b5c-327e6a0ba9a4_1_R/BasicLTI.xml I_02252cf0-e60c-3035-8b5c-327e6a0ba9a4_3_R/BasicLTI.xml I_02252cf0-e60c-3035-8b5c-327e6a0ba9a4_5_R/BasicLTI.xml I_02252cf0-e60c-3035-8b5c-327e6a0ba9a4_7_R/BasicLTI.xml I_02252cf0-e60c-3035-8b5c-327e6a0ba9a4_9_R/BasicLTI.xml I_02373678-d9f3-3a96-8c5c-eab8587ebc43_R/BasicLTI.xml I_0242a19b-ac18-34cc-9a49-8c1205820711_1_R/BasicLTI.xml I_0242a19b-ac18-34cc-9a49-8c1205820711_3_R/BasicLTI.xml I_0242a19b-ac18-34cc-9a49-8c1205820711_5_R/BasicLTI.xml I_0242a19b-ac18-34cc-9a49-8c1205820711_7_R/BasicLTI.xml I_0244dcea-66d4-34ef-b126-a67080790aa9_1_R/BasicLTI.xml I_02761161-3939-38ff-977f-e13294081393_1_R/BasicLTI.xml I_02761161-3939-38ff-977f-e13294081393_3_R/BasicLTI.xml I_02761161-3939-38ff-977f-e13294081393_5_R/BasicLTI.xml I_02761161-3939-38ff-977f-e13294081393_7_R/BasicLTI.xml I_029c470c-5df3-34d3-a7de-37f82281608a_1_R/BasicLTI.xml I_02a6d754-c725-3ae7-91ae-0ce6105035a8_1_R/BasicLTI.xml I_02b09316-82a6-3e25-8797-237ec7bdda9d_R/BasicLTI.xml I_02e1cb6e-8049-3aa7-b01e-11e14bf9d199_1_R/BasicLTI.xml I_02ff2f99-5cfe-3d9e-8f47-cfae7935bdc4_1_R/BasicLTI.xml I_03130a9f-6de4-3493-8892-ff6211110c34_1_R/BasicLTI.xml I_03130a9f-6de4-3493-8892-ff6211110c34_3_R/BasicLTI.xml I_034a8c98-5479-3a5c-9a14-232b7d37c189_1_R/BasicLTI.xml I_03545356-95e8-3bbe-9378-ca3c987819f5_1_R/BasicLTI.xml I_035f5bac-fcfa-39d9-98f8-d2a718f01a42_1_R/BasicLTI.xml I_03702864-ac20-3a17-aa4c-c48eaa00e664_1_R/BasicLTI.xml I_037bae30-53b2-3e6b-93c9-3b339b01f76d_R/BasicLTI.xml I_0392a1bd-a53f-36f7-b64d-959ca6ca99e1_1_R/BasicLTI.xml I_0392a1bd-a53f-36f7-b64d-959ca6ca99e1_3_R/BasicLTI.xml I_0397d761-8e36-35f2-826f-98c622ad8466_R/BasicLTI.xml I_0398a3f5-e66d-3df6-9543-4739c026dc86_1_R/BasicLTI.xml I_03a0b94f-2ea5-3039-9e53-e806cb33bfb3_1_R/BasicLTI.xml I_03ab5a4c-208f-3920-ad71-562345069c7e_1_R/BasicLTI.xml I_03b403ec-aec8-358d-bde8-f4ca1de4e205_1_R/BasicLTI.xml I_03b4cf8a-3a9c-3262-a318-be89af033483_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_11_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_1_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_3_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_5_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_7_R/BasicLTI.xml I_03b6d27c-17ce-3849-b3ee-78ba3cd03a26_9_R/BasicLTI.xml I_03f35725-4050-323b-b6a0-1b143300df9e_1_R/BasicLTI.xml I_03fed328-8c04-37d3-95a0-c13765e62c34_1_R/BasicLTI.xml I_0449fd07-4b02-3252-8eb9-b52b52ea288e_1_R/BasicLTI.xml I_046cf0dc-d1b0-3641-9e86-6cb7246849d4_R/BasicLTI.xml I_0477bce3-a424-35fc-a7fa-85cf7dd0842c_1_R/BasicLTI.xml I_0477c66d-95fd-3f17-a452-8f708cd6e5e4_1_R/BasicLTI.xml I_0481817b-d363-313c-8072-394747899cb8_R/BasicLTI.xml I_049396e3-64ca-3fd0-8d50-b21473473387_1_R/BasicLTI.xml I_04b08f30-2abb-3cc7-97d9-55abdab854ef_1_R/BasicLTI.xml I_04b2285f-e5c5-3004-9387-3dd584450dfc_R/BasicLTI.xml I_04b89841-019f-3a38-8823-a296fb944ee5_1_R/BasicLTI.xml I_04c86dd7-90f2-35ab-b040-c35b592951ff_R/BasicLTI.xml I_04df275d-7e7d-3aae-88ec-8321b57042ca_1_R/BasicLTI.xml I_04df275d-7e7d-3aae-88ec-8321b57042ca_3_R/BasicLTI.xml I_05511167-878f-31aa-a707-28ca3a732ff4_1_R/BasicLTI.xml I_055e332d-9893-310e-8cf3-9dfdefe95262_1_R/BasicLTI.xml I_05618b94-caa1-3e00-84f5-4815008bf00a_1_R/BasicLTI.xml I_05745501-b2e7-396a-8ec9-992988117082_1_R/BasicLTI.xml I_0579aea1-efc6-392e-84b4-83967dc42f58_1_R/BasicLTI.xml I_0579cd50-e92c-33ec-b598-a63a3407db5e_1_R/BasicLTI.xml I_0582252b-ae56-311a-98f0-1964893f10fe_1_R/BasicLTI.xml I_0587630d-331c-34ed-a486-fd82b0cd09f2_R/BasicLTI.xml I_05a08ecb-bf4d-3f01-9ce8-5ee00dead242_1_R/BasicLTI.xml I_05a7d09f-09a6-38d6-86f6-527c4bd74a12_1_R/BasicLTI.xml I_05c63987-3a53-3ba9-a274-2ddddab9257a_R/BasicLTI.xml I_05cc2bdb-9cb3-3227-80ea-4a4a41d0e45c_1_R/BasicLTI.xml I_05df3374-f320-3f26-9a49-b5d7b45d5f78_1_R/BasicLTI.xml I_05faa8a4-5ff4-3e44-924f-f742cc16273c_R/BasicLTI.xml I_05fed691-45e1-38d5-9ff2-1c3376a17466_1_R/BasicLTI.xml I_05ff21ab-63c4-339a-aafa-8ad9b8d1c64b_1_R/BasicLTI.xml I_0608f775-5d21-3551-86aa-d269bba4e0fd_R/BasicLTI.xml I_061f824b-168e-3876-ba14-5128afbe4a14_1_R/BasicLTI.xml I_062eee7e-fdb2-324f-8644-b2c9d3b9d8b1_1_R/BasicLTI.xml I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_1_R/BasicLTI.xml I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_3_R/BasicLTI.xml I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_5_R/BasicLTI.xml I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_7_R/BasicLTI.xml I_0656c249-1cc5-3be4-afa5-06ee87c05a5d_9_R/BasicLTI.xml I_068975c4-867e-35a2-99a6-39df9c59ed16_R/BasicLTI.xml I_06b41710-e759-3af3-b222-4acfd9587ce0_1_R/BasicLTI.xml I_06b785d5-6999-3c77-8d33-95f78ddec25c_1_R/BasicLTI.xml I_06bdb59e-a058-36d0-b1ea-cee6e9f926b8_R/BasicLTI.xml I_06fa6a93-638d-3ad6-af51-b750d836325b_1_R/BasicLTI.xml I_06fd1751-da11-32d4-9cd8-1d69ce3ac463_1_R/BasicLTI.xml I_0700e554-dda9-3a9f-aad3-36b7362d1f5a_1_R/BasicLTI.xml I_071430ad-8d4d-3b41-957e-174010e99df3_1_R/BasicLTI.xml I_072d9e94-7894-3239-86bb-cda1ed4d2235_R/BasicLTI.xml I_072fb7f0-a4a6-3315-8852-f90aae28539c_1_R/BasicLTI.xml I_0737dd52-fb1b-3ffc-b1fb-60dcdd702cae_1_R/BasicLTI.xml I_074d378e-83f8-3694-9730-aa6f891cec79_1_R/BasicLTI.xml I_0757f450-893d-3615-abed-d893f2132619_R/BasicLTI.xml I_075fc3c4-4505-3886-856d-a38188d56213_1_R/BasicLTI.xml I_07667641-56f6-3c04-b9e4-99e58c6a1bb3_R/BasicLTI.xml I_076ba0fe-6262-3d7b-8cea-2c9d37ae07c4_1_R/BasicLTI.xml I_076d1e4c-706f-32a3-aafe-5ecdbf7b927c_R/BasicLTI.xml I_0773961b-c356-31f9-825c-54217c0fc0b9_1_R/BasicLTI.xml I_07915af5-b687-3212-b001-016cd9143cdc_1_R/BasicLTI.xml I_07915af5-b687-3212-b001-016cd9143cdc_3_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_11_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_1_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_3_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_5_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_7_R/BasicLTI.xml I_07a31fb0-910d-34dc-bb86-4a8038f11737_9_R/BasicLTI.xml I_07a5600b-7fb6-36e6-aa29-1ccdc3846387_R/BasicLTI.xml I_07b41463-4f04-33d6-888d-b4e36112fd01_R/BasicLTI.xml I_07b8278d-36bd-32ab-a90a-40f0495b3947_1_R/BasicLTI.xml I_07c15778-ca5a-36dc-939b-61b1c77039ef_R/BasicLTI.xml I_07c20a9c-9902-3c95-9bd9-00d3d2f8a355_1_R/BasicLTI.xml I_07c60fe4-e13f-3cd2-a567-b7d2323cc1c4_R/BasicLTI.xml I_07fc0b2e-4420-3a5e-9a02-3de81c93419c_1_R/BasicLTI.xml I_081aa0ae-d050-3d6e-90ac-b896c4c18f6c_1_R/BasicLTI.xml I_0823b9bd-0bd2-3382-af89-06559d24aef6_1_R/BasicLTI.xml I_0826d052-5a07-38e4-b1ce-329fe8545a06_R/BasicLTI.xml I_08563f11-ab9b-3abb-9967-55695dbc5831_1_R/BasicLTI.xml I_08579d15-4387-344b-bd13-d6f7a2a6a4b9_R/BasicLTI.xml I_0866e965-1fc7-314b-bb38-c4b18d5dbd32_1_R/BasicLTI.xml I_086d350d-8fca-376d-8e7c-45a75720c38a_1_R/BasicLTI.xml I_086d7cc9-bb43-3b88-b27f-4ac2e23e767c_R/BasicLTI.xml I_08725500-8023-389c-b0c2-94a603ef0f17_R/BasicLTI.xml I_0876854b-1b70-3a72-a09c-1fc7091cd22c_R/BasicLTI.xml I_087e9b5f-18d2-31bd-8923-7a6fafe895e5_1_R/BasicLTI.xml I_088ea942-424b-3de4-821f-5eb8a99e8686_1_R/BasicLTI.xml I_0895967b-86d5-35f5-9844-9e3b565e7d97_R/BasicLTI.xml I_08aa42a6-99a3-3f1e-9e19-a1a7ce682d61_1_R/BasicLTI.xml I_08aff233-6871-3ecd-9552-001d0fba8c0e_1_R/BasicLTI.xml I_08aff233-6871-3ecd-9552-001d0fba8c0e_3_R/BasicLTI.xml I_08aff233-6871-3ecd-9552-001d0fba8c0e_5_R/BasicLTI.xml I_08ea2a4f-25f7-30fc-8bed-02824bbd1dcd_1_R/BasicLTI.xml I_08ec0359-4508-3b02-80fe-918d7a41589f_1_R/BasicLTI.xml I_090a8dd1-8dcf-3fcc-966b-7f119675a2c4_1_R/BasicLTI.xml I_09124cfc-852d-334f-8ef5-5cc3636675c2_1_R/BasicLTI.xml I_095f9432-5f2b-3481-981f-01530f35a731_1_R/BasicLTI.xml I_09923e5f-e976-3740-90a9-b8d85b66d21c_1_R/BasicLTI.xml I_09b5e491-687d-351d-826e-a40dd878fc84_1_R/BasicLTI.xml I_09c257b0-44a7-389e-8336-8237fe5d9870_R/BasicLTI.xml I_09ee11b2-5b8b-3066-b755-c757770ca411_1_R/BasicLTI.xml I_0a096c90-6fea-3764-a404-e38d00495908_R/BasicLTI.xml I_0a10650f-ee5d-31f5-ac95-c7e09ddbd4ed_1_R/BasicLTI.xml I_0a10650f-ee5d-31f5-ac95-c7e09ddbd4ed_3_R/BasicLTI.xml I_0a20c317-fe77-359d-bc43-6914fdfab54f_R/BasicLTI.xml I_0a25e329-7d08-37e1-84a9-89ceeca596f0_1_R/BasicLTI.xml I_0a395cf9-1ced-3ecf-9b0a-a3915ea2cb1a_1_R/BasicLTI.xml I_0a6f696f-e24a-333c-bc9e-f7d4ada1b8e4_1_R/BasicLTI.xml I_0aa862a0-021a-3d40-86ba-8199c8d2a0ef_1_R/BasicLTI.xml I_0aaa4fa4-11f8-3e20-a04d-8fa7c76d2ba6_1_R/BasicLTI.xml I_0aaf1fb3-0ad6-34d0-8c97-57709fb66f8a_1_R/BasicLTI.xml I_0ab24b46-35d6-332f-8150-878a4eeb5243_1_R/BasicLTI.xml I_0ab3a46b-8e01-390f-845b-d41f2bf74e34_R/BasicLTI.xml I_0ac2df55-f9c5-3ce8-9096-ec5621f22e6e_1_R/BasicLTI.xml I_0ac2df55-f9c5-3ce8-9096-ec5621f22e6e_3_R/BasicLTI.xml I_0ac2df55-f9c5-3ce8-9096-ec5621f22e6e_5_R/BasicLTI.xml I_0adb4485-d30f-3320-86fa-d8c11755b3bd_1_R/BasicLTI.xml I_0b0d16b5-0965-3f56-9002-7685f4b08aae_1_R/BasicLTI.xml I_0b0d16b5-0965-3f56-9002-7685f4b08aae_3_R/BasicLTI.xml I_0b0d16b5-0965-3f56-9002-7685f4b08aae_5_R/BasicLTI.xml I_0b0d16b5-0965-3f56-9002-7685f4b08aae_7_R/BasicLTI.xml I_0b0d16b5-0965-3f56-9002-7685f4b08aae_9_R/BasicLTI.xml I_0b27c147-7c59-342d-958b-322e421b604c_1_R/BasicLTI.xml I_0b2b3892-fc5a-3316-8374-956617839ddd_R/BasicLTI.xml I_0b479db3-f0f4-3b0f-9c64-9eae8971a499_R/BasicLTI.xml I_0b7da9d4-500d-3ce6-bc94-f7419156c36c_1_R/BasicLTI.xml I_0b8ae1af-b6e3-3ef2-8b86-aa0fdc3e27f3_1_R/BasicLTI.xml I_0b922255-8610-3321-b07b-9c22e72a1b51_R/BasicLTI.xml I_0ba123b5-78ba-33b7-a47f-14479f86e214_1_R/BasicLTI.xml I_0ba5b594-a5d5-37ac-8cc7-646ab7320c24_R/BasicLTI.xml I_0bb1ecd1-9102-32c9-81e1-b93627d0290b_1_R/BasicLTI.xml I_0bbb56e8-81a2-3039-bfd7-6f979841dceb_1_R/BasicLTI.xml I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_1_R/BasicLTI.xml I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_3_R/BasicLTI.xml I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_5_R/BasicLTI.xml I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_7_R/BasicLTI.xml I_0bc7efe2-1914-383a-b6d3-8cac06b1cbe7_9_R/BasicLTI.xml I_0bf4c73f-a895-31fc-80d8-759a4e951c44_1_R/BasicLTI.xml I_0bf6a36d-3eb1-3093-907c-453ee42a95a1_1_R/BasicLTI.xml I_0c3a99cf-5828-346d-91b8-d6c1b63e88f9_R/BasicLTI.xml I_0c54f164-7ce4-35cd-bfcc-b3b9af6d850c_1_R/BasicLTI.xml I_0c86be10-43ce-3c66-85af-3a0607a099db_1_R/BasicLTI.xml I_0c96cf1c-e454-3bd6-9649-22c0983f26be_R/BasicLTI.xml I_0c9ebe24-4b21-32cb-a6d2-e1718ca4a7ca_1_R/BasicLTI.xml I_0ca0e1f1-656d-3e6d-b860-8cf11c5b97f0_1_R/BasicLTI.xml I_0ca10156-76f2-302b-8713-5bca4ee4d823_1_R/BasicLTI.xml I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_1_R/BasicLTI.xml I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_3_R/BasicLTI.xml I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_5_R/BasicLTI.xml I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_7_R/BasicLTI.xml I_0cb55a55-ebc4-3b8e-a0e8-7b0e81870501_9_R/BasicLTI.xml I_0cbc7f75-1ee4-3bdf-8742-8925de8eac8b_1_R/BasicLTI.xml I_0cc07746-3a01-3515-8cfb-409e41da3948_R/BasicLTI.xml I_0cce9f20-914f-3fe9-ac1e-fb603b3ad2a4_1_R/BasicLTI.xml I_0ce7c957-5d1c-38b9-921d-6f2fff44894f_1_R/BasicLTI.xml I_0ce9f976-4b6c-3faa-9ce8-434cecde3b70_R/BasicLTI.xml I_0d2bf716-d25f-3eef-bfeb-82a5344c85ff_R/BasicLTI.xml I_0d5087d1-e402-3f36-83a9-9f7123c8885b_1_R/BasicLTI.xml I_0d533310-2623-334f-82c8-965d1e5e9403_1_R/BasicLTI.xml I_0d533310-2623-334f-82c8-965d1e5e9403_3_R/BasicLTI.xml I_0d533310-2623-334f-82c8-965d1e5e9403_5_R/BasicLTI.xml I_0d687781-ee94-348c-b2ee-6cdd981f48bc_1_R/BasicLTI.xml I_0d687781-ee94-348c-b2ee-6cdd981f48bc_3_R/BasicLTI.xml I_0d6934cf-f9b8-368e-8e09-a192fe64e110_1_R/BasicLTI.xml I_0d6afb97-9591-3837-a64d-9cba278b3871_R/BasicLTI.xml I_0d71e764-8733-3a67-a0fd-ac49c1af8912_R/BasicLTI.xml I_0d742d17-c701-37f4-bd48-7a6be8cc81f0_1_R/BasicLTI.xml I_0d742d17-c701-37f4-bd48-7a6be8cc81f0_3_R/BasicLTI.xml I_0d85c9ab-c4a6-3691-a219-a87c196f6667_1_R/BasicLTI.xml I_0d864beb-305f-3bff-a920-04c177683c55_R/BasicLTI.xml I_0d886136-ff62-344f-833e-d6a1fd0f47dc_R/BasicLTI.xml I_0da1da45-fb3b-37fb-af28-cb57f726951c_R/BasicLTI.xml I_0da6bc94-18e5-3c72-8f01-26190aa75c25_R/BasicLTI.xml I_0dc488e4-a6a8-31d2-8325-3a00b1fedf47_R/BasicLTI.xml I_0dc7272e-b334-3491-9077-1dbe9b94d266_1_R/BasicLTI.xml I_0dc74710-6641-3ee6-a54c-34aeb5f2a49c_1_R/BasicLTI.xml I_0de01d62-c9f7-30b9-b59d-aa328df35215_1_R/BasicLTI.xml I_0deabf46-f1c2-366f-8462-5200c1dc9f02_1_R/BasicLTI.xml I_0e01d66a-f838-392e-b75e-37f6ea9c7943_1_R/BasicLTI.xml I_0e05be72-f01c-308f-87c0-9c56a6465876_1_R/BasicLTI.xml I_0e05be72-f01c-308f-87c0-9c56a6465876_3_R/BasicLTI.xml I_0e05c27e-a19f-313d-8f88-554c0459405b_R/BasicLTI.xml I_0e06cf6d-e557-3e26-a599-dc603313097a_R/BasicLTI.xml I_0e1d80f4-0a77-32e9-8120-96e4153be47e_1_R/BasicLTI.xml I_0e20b3fb-ec38-3a17-97a7-8addbfd7deb0_1_R/BasicLTI.xml I_0e20b3fb-ec38-3a17-97a7-8addbfd7deb0_3_R/BasicLTI.xml I_0e20b3fb-ec38-3a17-97a7-8addbfd7deb0_5_R/BasicLTI.xml I_0e20b3fb-ec38-3a17-97a7-8addbfd7deb0_7_R/BasicLTI.xml I_0e4be8ac-0bb2-36a7-810e-d63a87c0101d_1_R/BasicLTI.xml I_0e5848a6-844e-3ce0-bcf3-e323d968b105_1_R/BasicLTI.xml I_0e5cae0d-b50f-36e2-95fc-bb41df2e989d_1_R/BasicLTI.xml I_0e832d51-e16c-3e52-80fd-434faa701f6e_1_R/BasicLTI.xml I_0e832d51-e16c-3e52-80fd-434faa701f6e_3_R/BasicLTI.xml I_0e95e50c-2acf-3737-bcb2-b4dfaf2b6c9a_1_R/BasicLTI.xml I_0ea21a8f-5715-3fbb-80a1-cd2fa68993f4_1_R/BasicLTI.xml I_0ea5b963-3589-3703-8e30-f15e7c545f63_R/BasicLTI.xml I_0ebdedd8-86a6-314a-9f0b-6b1360e29730_1_R/BasicLTI.xml I_0ec0c9ad-db72-39c5-ae2a-22474c0e42ae_1_R/BasicLTI.xml I_0eed1b08-9930-3d7a-941b-e873e039e4fc_R/BasicLTI.xml I_0ef85f19-7e92-39db-933a-5f600a2f522e_1_R/BasicLTI.xml I_0f147667-5417-3181-8ada-7cd04f24b088_1_R/BasicLTI.xml I_0f17fd4e-f481-3b61-8cb8-8ce5e96efe7d_1_R/BasicLTI.xml I_0f17fd4e-f481-3b61-8cb8-8ce5e96efe7d_3_R/BasicLTI.xml I_0f32137d-ee24-3ac3-ab67-5ae7c626c203_R/BasicLTI.xml I_0f3287bc-8f12-3e4f-9481-e0e5c4eb7b19_1_R/BasicLTI.xml I_0f59a764-f7b8-3101-a959-efea36fb29ff_R/BasicLTI.xml I_0f639b70-9224-3cfa-9998-767feefd3535_1_R/BasicLTI.xml I_0f761d55-53ea-3a16-b2c9-86ae508553c2_R/BasicLTI.xml I_0f865712-6443-3119-9e75-d8496a264f25_1_R/BasicLTI.xml I_0f883531-9ab6-3788-98e8-9c8562bc8234_R/BasicLTI.xml I_0f9b37d0-d8fe-3c73-9967-666447a88aad_R/BasicLTI.xml I_0fa6c4a5-6122-3557-b557-39b80b3896c5_R/BasicLTI.xml I_0fb0eead-8763-3d66-a2fc-655c3add4d71_1_R/BasicLTI.xml I_0fee762a-cfa0-31d7-a01c-624150f754cc_1_R/BasicLTI.xml I_103ef125-fdbd-31ba-a4e9-64d51a66c7a1_R/BasicLTI.xml I_10431579-2d93-3648-bb10-da6137388869_1_R/BasicLTI.xml I_1058cc64-9a1e-3923-8363-ee184aa5adcd_R/BasicLTI.xml I_10a97f8b-f61f-3aa8-8611-32d7b962de8d_1_R/BasicLTI.xml I_10b2697d-1bd2-3acc-a990-cd6121dc95a1_1_R/BasicLTI.xml I_10c54836-5a77-3a73-a35b-b59a50541e61_1_R/BasicLTI.xml I_10d17994-e1c5-36ce-b386-59b9c5e81bde_1_R/BasicLTI.xml I_10de7e59-f8a9-33fd-b17c-e19c3f0bcd47_1_R/BasicLTI.xml I_10eb12a7-b2a2-3a9b-8452-33e9a6389b38_1_R/BasicLTI.xml I_10eb12a7-b2a2-3a9b-8452-33e9a6389b38_3_R/BasicLTI.xml I_10ec24dc-4e91-37a5-88d6-e4eed1f44dac_1_R/BasicLTI.xml I_111e8223-33b7-3103-b8f6-82dc6f3b9b33_1_R/BasicLTI.xml I_1130f2c6-96fc-317b-9ff6-d7e74c59467d_1_R/BasicLTI.xml I_1156a4f8-17b9-37bf-a36f-a26c19226192_R/BasicLTI.xml I_1168c8ca-5d78-3d63-8452-d8717961fda9_1_R/BasicLTI.xml I_1168c8ca-5d78-3d63-8452-d8717961fda9_3_R/BasicLTI.xml I_1178e50b-cd0d-34f6-9919-36b736f78b16_1_R/BasicLTI.xml I_1178e50b-cd0d-34f6-9919-36b736f78b16_3_R/BasicLTI.xml I_11881a8c-3528-3131-bcf1-6980550efd00_1_R/BasicLTI.xml I_11881a8c-3528-3131-bcf1-6980550efd00_3_R/BasicLTI.xml I_11881a8c-3528-3131-bcf1-6980550efd00_5_R/BasicLTI.xml I_1197fe88-e4f5-3497-800b-297af1e03cd1_1_R/BasicLTI.xml I_11af85d9-0747-3071-9e1e-31c3360cf022_1_R/BasicLTI.xml I_11b1cfbd-3abb-3fe7-942a-de60f25d1b09_1_R/BasicLTI.xml I_11f05baf-0372-3bc1-9613-02d8d989997f_1_R/BasicLTI.xml I_11fb88e8-71af-37bf-8019-4a5a3faac29d_R/BasicLTI.xml I_11fea1a1-f7af-3317-8cc6-96c454a69aaa_1_R/BasicLTI.xml I_11fea1a1-f7af-3317-8cc6-96c454a69aaa_3_R/BasicLTI.xml I_11fea1a1-f7af-3317-8cc6-96c454a69aaa_5_R/BasicLTI.xml I_1208008e-3585-370b-a1ae-d9392097c527_R/BasicLTI.xml I_1213c17c-d669-35e5-a443-a890e206cd1c_1_R/BasicLTI.xml I_123a8f54-bc47-3165-8f8c-69c73ccee912_R/BasicLTI.xml I_1243f987-dea0-3077-9aeb-1cae3a07953f_R/BasicLTI.xml I_125007be-797d-3b3b-b76f-5238e6805940_1_R/BasicLTI.xml I_12527e92-cef9-3367-8525-b736a1a6883c_1_R/BasicLTI.xml I_125ed029-abd8-3f7d-98b3-b8875d043e2d_1_R/BasicLTI.xml I_1271ff69-8403-33e7-be64-5d45ff8daf0f_R/BasicLTI.xml I_12757c0b-1bee-382a-9370-69075a3765bd_1_R/BasicLTI.xml I_127656ec-2b20-38bc-8c86-d6efb36846e7_1_R/BasicLTI.xml I_12833ef9-f07a-3026-99cc-f1352803f568_R/BasicLTI.xml I_12becf19-fa85-3317-97c1-6b50e9b3fb05_1_R/BasicLTI.xml I_12c74576-a4aa-387e-8536-d20372204c59_R/BasicLTI.xml I_12cf9175-4b73-38fb-ab57-491e175c4b6d_1_R/BasicLTI.xml I_12de6a42-8e6b-3b83-9cb1-a4d69f2d11da_1_R/BasicLTI.xml I_12f26ffd-fd3e-3e0f-bf38-93aa973f2e72_R/BasicLTI.xml I_12f2f069-2a87-3999-9e8a-f27c4a5f2ba5_R/BasicLTI.xml I_1307096b-a79f-3e2a-8295-8068494416da_1_R/BasicLTI.xml I_13088ab7-89a1-3bf7-a49c-3dbad49b570a_R/BasicLTI.xml I_13149fb0-dc9e-3b3f-b76c-5392470135d6_1_R/BasicLTI.xml I_131b21d4-db5a-3ef8-8da5-b7e529556115_R/BasicLTI.xml I_13283aec-e78c-3fe7-a761-786f6c14319f_1_R/BasicLTI.xml I_132930f2-6036-30d9-a3b4-b37540c362ca_R/BasicLTI.xml I_132a4d8b-be9b-3a7c-b7a8-df4a16593d47_1_R/BasicLTI.xml I_132cad96-6de4-3df9-bd89-97547fd55e18_1_R/BasicLTI.xml I_132cad96-6de4-3df9-bd89-97547fd55e18_3_R/BasicLTI.xml I_1355c847-e2ef-33a2-bd80-43ae4adba597_1_R/BasicLTI.xml I_13588034-af53-3681-86b0-4e70d93e058c_1_R/BasicLTI.xml I_136e992d-58d5-386f-aad7-09c3bbe8a996_1_R/BasicLTI.xml I_137a5322-0c07-3a42-95c3-0d95cf2772c9_1_R/BasicLTI.xml I_138b49c3-b127-3627-b7f8-b655cd40af7d_1_R/BasicLTI.xml I_138b49c3-b127-3627-b7f8-b655cd40af7d_3_R/BasicLTI.xml I_138c3761-7e2b-3aa0-a85d-4a8c5461234d_1_R/BasicLTI.xml I_13bb33d2-9d85-37f6-b94f-6317cb9b679a_R/BasicLTI.xml I_14244471-8892-3b79-a38f-0654a4b19c36_R/BasicLTI.xml I_1463a4a1-349d-3da9-90f1-ba9a90558a87_R/BasicLTI.xml I_146a38c2-a516-3e02-b3a4-37e08f320004_R/BasicLTI.xml I_1496b1c3-b4fa-38a2-92bc-8ac89464e682_1_R/BasicLTI.xml I_149b9187-e7bb-3038-a744-9faecfb8cdd2_1_R/BasicLTI.xml I_14a29a86-fa83-3b61-a57b-141ee8fb2b2c_1_R/BasicLTI.xml I_14c792f0-c34a-3d16-ae05-60569d04d73c_1_R/BasicLTI.xml I_14d6252c-02e2-3946-933f-600e44ad494f_R/BasicLTI.xml I_14dea7e9-218b-36ca-8b48-0cb8c6e08d56_1_R/BasicLTI.xml I_14e8c29d-5f07-3180-aa1f-899b87ab5fbf_1_R/BasicLTI.xml I_14ff098e-6f94-3b9b-9d37-daa216cce580_1_R/BasicLTI.xml I_150159b7-96d0-3495-a6f9-b65f84940270_R/BasicLTI.xml I_150a2b64-8325-3180-ae33-c8c1c413a6ca_R/BasicLTI.xml I_151a75ee-90da-34c4-b568-735abf32b4c8_1_R/BasicLTI.xml I_15201ed8-eb57-364b-8ee3-4f2bff93e306_1_R/BasicLTI.xml I_154cac42-f94a-3c01-b9e1-579c231a8aed_R/BasicLTI.xml I_15500ecc-93d2-3439-89c7-56a7eee845c3_R/BasicLTI.xml I_158338f5-caee-326c-8ff6-34f8c9445e41_R/BasicLTI.xml I_158f9b94-1f73-3229-b38e-a07be65b0da6_R/BasicLTI.xml I_159610e5-ad9a-3b4f-89db-7e8cf3c3e4ad_R/BasicLTI.xml I_15bf42e2-0672-3a74-9fc0-f3801ebe7206_1_R/BasicLTI.xml I_15bf42e2-0672-3a74-9fc0-f3801ebe7206_3_R/BasicLTI.xml I_15cb4f5e-585f-34ce-adaa-d2f5467e0eb6_R/BasicLTI.xml I_15cf6a7e-6060-3d95-8ec9-1251ab66350b_1_R/BasicLTI.xml I_15d5ad2c-9dbe-3c27-b482-dbdc6139d044_1_R/BasicLTI.xml I_15daf3f2-ea24-3c86-a8ab-d2aed3f65632_1_R/BasicLTI.xml I_15f4773f-9a8a-3864-8f1b-39d43ef0b2d8_1_R/BasicLTI.xml I_15f7303f-a20c-3ce5-9dd8-866ed14cb070_1_R/BasicLTI.xml I_15f7303f-a20c-3ce5-9dd8-866ed14cb070_3_R/BasicLTI.xml I_15f7303f-a20c-3ce5-9dd8-866ed14cb070_5_R/BasicLTI.xml I_1603bf58-9fcc-3047-aaf6-a081e446c242_1_R/BasicLTI.xml I_1603bf58-9fcc-3047-aaf6-a081e446c242_3_R/BasicLTI.xml I_1609f6ed-30d8-35ec-8fe7-40a410957693_1_R/BasicLTI.xml I_160f1aaf-d8b5-3686-8334-320927ad7ead_R/BasicLTI.xml I_16129f5e-830b-3871-9c7f-2ede90b521f9_1_R/BasicLTI.xml I_16187817-0801-3ab1-a911-0c2736f65b41_1_R/BasicLTI.xml I_16286976-c51d-382e-919b-2e37452dcf67_1_R/BasicLTI.xml I_16286976-c51d-382e-919b-2e37452dcf67_3_R/BasicLTI.xml I_1630af1b-8b8c-3bae-82bb-bc6018bd8f95_1_R/BasicLTI.xml I_1630af1b-8b8c-3bae-82bb-bc6018bd8f95_3_R/BasicLTI.xml I_163cb96b-a30e-382c-8e96-64a3b47c42b0_R/BasicLTI.xml I_16429cbe-b6c3-3249-8ff2-284c93be9698_R/BasicLTI.xml I_1658d7c0-1320-3b23-8cb6-cd6b779b5488_R/BasicLTI.xml I_166b484c-e8a6-3b13-a396-3fa8d2ca6574_1_R/BasicLTI.xml I_1675a7d6-3112-3079-8d2f-36c30eb819dc_R/BasicLTI.xml I_16877f3e-7619-378f-8c79-986f9b652ef6_R/BasicLTI.xml I_1687f338-28ee-3887-950b-d5343a6c2935_1_R/BasicLTI.xml I_16a7a20c-be4e-377a-aff9-2f31ad54ba76_R/BasicLTI.xml I_16aef35b-523a-3113-91cc-334a99e6b85a_R/BasicLTI.xml I_16c35757-b870-3b1d-8fa2-21d4b89645c0_1_R/BasicLTI.xml I_16d1222f-77b2-36fb-ba54-a640969bcf27_1_R/BasicLTI.xml I_16d89b40-06fa-357c-bde9-4c060b2e881c_1_R/BasicLTI.xml I_171468da-0808-3d4e-a000-37f867dca29c_1_R/BasicLTI.xml I_17213b99-fd11-38d8-a972-1858b4158106_1_R/BasicLTI.xml I_172943cf-341a-3d34-bc08-5e4120f80863_1_R/BasicLTI.xml I_17ce3089-8ade-3510-9eeb-8a19d07eed2f_R/BasicLTI.xml I_17f6f158-032d-3c10-b037-f282b1a79f8d_R/BasicLTI.xml I_17f8841a-cd67-345b-92dd-e5c7386a1f7e_1_R/BasicLTI.xml I_17fd09f8-c54c-3e81-9365-c2c874444dec_1_R/BasicLTI.xml I_183b47b4-2bef-3d8e-9983-48c66071df65_1_R/BasicLTI.xml I_184205b7-0493-34f8-8055-ad7b486f2796_R/BasicLTI.xml I_187408d0-9b1b-3c1a-89b7-6a454a419080_1_R/BasicLTI.xml I_187408d0-9b1b-3c1a-89b7-6a454a419080_3_R/BasicLTI.xml I_187408d0-9b1b-3c1a-89b7-6a454a419080_5_R/BasicLTI.xml I_187408d0-9b1b-3c1a-89b7-6a454a419080_7_R/BasicLTI.xml I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_1_R/BasicLTI.xml I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_3_R/BasicLTI.xml I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_5_R/BasicLTI.xml I_1876c17b-6d38-3b39-aaae-e7a84a4e698c_7_R/BasicLTI.xml I_18888b3b-5495-3e7e-aa1d-36bcf456c6e7_1_R/BasicLTI.xml I_18888b3b-5495-3e7e-aa1d-36bcf456c6e7_3_R/BasicLTI.xml I_18b383e4-d621-3630-9384-6d34bb5d6098_R/BasicLTI.xml I_18c77d48-71ec-36e2-95fc-6d08d74a0293_1_R/BasicLTI.xml I_18c7d81b-ea69-3e28-87e1-007d3198d9fc_1_R/BasicLTI.xml I_18c7d81b-ea69-3e28-87e1-007d3198d9fc_3_R/BasicLTI.xml I_18c7d81b-ea69-3e28-87e1-007d3198d9fc_5_R/BasicLTI.xml I_18c7d81b-ea69-3e28-87e1-007d3198d9fc_7_R/BasicLTI.xml I_18dd6e01-a055-35cc-9d1f-8344c92feff7_1_R/BasicLTI.xml I_18dfb3af-fe79-365e-acbd-c11c891af004_R/BasicLTI.xml I_18e06ea8-1520-3253-a4d0-2ea315caddcc_1_R/BasicLTI.xml I_1901038c-5118-3697-a184-a479d540b679_R/BasicLTI.xml I_1909dc0e-7d76-3f52-8a27-fae0af7c67b3_1_R/BasicLTI.xml I_19257721-8a03-3140-8644-eb3991c36eba_1_R/BasicLTI.xml I_193fd6be-6446-3088-a7fa-bf79ae739687_R/BasicLTI.xml I_19454d32-156e-391e-ac1c-04c05307533d_1_R/BasicLTI.xml I_19537822-68e6-3b02-87a2-e19f9a06432d_R/BasicLTI.xml I_1959f942-ea73-38ae-a198-309eedea61ba_1_R/BasicLTI.xml I_1959f942-ea73-38ae-a198-309eedea61ba_3_R/BasicLTI.xml I_1959f942-ea73-38ae-a198-309eedea61ba_5_R/BasicLTI.xml I_1960306e-cf9d-3216-be31-051390209e8a_1_R/BasicLTI.xml I_1960306e-cf9d-3216-be31-051390209e8a_3_R/BasicLTI.xml I_1960306e-cf9d-3216-be31-051390209e8a_5_R/BasicLTI.xml I_1960306e-cf9d-3216-be31-051390209e8a_7_R/BasicLTI.xml I_1960306e-cf9d-3216-be31-051390209e8a_9_R/BasicLTI.xml I_197492d5-8463-3795-80d2-cd0704d4dd4e_1_R/BasicLTI.xml I_19f902ef-ef9d-3e2c-b0cb-b6929fa4feea_R/BasicLTI.xml I_1a1149d1-c9f2-33fb-b351-f20d6d05ad1e_1_R/BasicLTI.xml I_1a1a3f70-0d64-3665-9718-669e8de4dea5_R/BasicLTI.xml I_1a24f5d3-7598-3c59-933d-0f41747aee93_1_R/BasicLTI.xml I_1a37d9ec-1bf0-3fba-b795-97bf39ecd87d_1_R/BasicLTI.xml I_1a406ff0-d5c2-31f5-b788-88bedc3cd58d_R/BasicLTI.xml I_1a43ca04-c68d-3f5d-bf9c-5ec03ca9436c_1_R/BasicLTI.xml I_1a54e228-b0a6-3d26-85ae-8c8f87efbd09_1_R/BasicLTI.xml I_1a6c8832-7116-3e8a-9ed0-35e67fb430bd_1_R/BasicLTI.xml I_1a7e35d0-1b70-34c1-876a-7e29412e8329_1_R/BasicLTI.xml I_1aa5e528-2963-3b1b-ab2b-78dd22925276_1_R/BasicLTI.xml I_1ac0cd1f-87a9-3543-8fd5-05b5ea48a953_1_R/BasicLTI.xml I_1ac5a2b0-02f1-39db-a5dd-a92bfccea84f_R/BasicLTI.xml I_1acab93c-5709-38c9-ab19-f9be775ed184_R/BasicLTI.xml I_1ad23e30-d182-37d5-a1a3-815719b115a2_1_R/BasicLTI.xml I_1ad23e30-d182-37d5-a1a3-815719b115a2_3_R/BasicLTI.xml I_1ad23e30-d182-37d5-a1a3-815719b115a2_5_R/BasicLTI.xml I_1ad23e30-d182-37d5-a1a3-815719b115a2_7_R/BasicLTI.xml I_1add14bc-8137-3b66-a149-90f1cc4d217f_R/BasicLTI.xml I_1b0b2d40-8219-3e38-a4d6-9571c6d0a2ed_1_R/BasicLTI.xml I_1b33fc29-546b-3ee1-92b3-64a24b954b17_R/BasicLTI.xml I_1b49076d-31ff-34d1-8543-676b6f92168f_R/BasicLTI.xml I_1b4d5dd1-9da0-34cf-ba1f-8727eb69fcb2_R/BasicLTI.xml I_1b737501-fd6c-32e6-8283-2e488565a125_R/BasicLTI.xml I_1b7ef00e-46ac-39f1-bbca-00328ba5f542_1_R/BasicLTI.xml I_1b9419e5-cce6-36ac-8a25-4cd6f545dfdf_R/BasicLTI.xml I_1bd09113-5554-35e3-b87b-6f256b49fcaa_R/BasicLTI.xml I_1bed0fe5-d927-3c60-8dee-ef69187f1e5d_R/BasicLTI.xml I_1c046a69-03f9-35fd-8aa0-f9819ef47e81_1_R/BasicLTI.xml I_1c062472-d862-3563-b967-7a2a9d6f4318_1_R/BasicLTI.xml I_1c2d02ab-4eba-3c4f-86b1-98ce07f4fc66_R/BasicLTI.xml I_1c359bf6-6750-3443-9b58-7b0ddbe66833_R/BasicLTI.xml I_1c384084-ff8a-3a9d-bbf6-ecff2be717ba_1_R/BasicLTI.xml I_1c486bf3-68e1-33ee-86fe-66486d38ab19_1_R/BasicLTI.xml I_1c654c92-63b0-3681-8e6f-d3a0a3becaa0_1_R/BasicLTI.xml I_1c66f082-e081-3149-b083-3a8abcee9796_R/BasicLTI.xml I_1c788202-d61f-3594-9b35-92818ae8aaa1_1_R/BasicLTI.xml I_1c8103d8-1c93-366d-8eb3-b94cadb19635_R/BasicLTI.xml I_1c936dd2-257f-3a98-a580-7b08cd688c40_1_R/BasicLTI.xml I_1c936dd2-257f-3a98-a580-7b08cd688c40_3_R/BasicLTI.xml I_1c936dd2-257f-3a98-a580-7b08cd688c40_5_R/BasicLTI.xml I_1c936dd2-257f-3a98-a580-7b08cd688c40_7_R/BasicLTI.xml I_1c936dd2-257f-3a98-a580-7b08cd688c40_9_R/BasicLTI.xml I_1ca088ca-db04-3f01-b177-38005748bb26_R/BasicLTI.xml I_1cadbc47-f831-37f0-855e-93fa45c23fe3_R/BasicLTI.xml I_1cc5ac46-ca00-3330-8d22-54993fca280e_1_R/BasicLTI.xml I_1ccdeccc-98fb-397c-86eb-1891ca748cd7_1_R/BasicLTI.xml I_1cdb2cd0-745b-34e1-b950-0e7aaa31bcdb_R/BasicLTI.xml I_1ce2b998-0001-3f4c-86fc-ed06f76915ee_1_R/BasicLTI.xml I_1ce3c429-eeae-3130-8dd0-668497aec53f_R/BasicLTI.xml I_1ce4fd98-1c15-332d-94c2-00f1bf6319c3_R/BasicLTI.xml I_1d05e17d-b134-34d7-adb6-e1635dc9715b_1_R/BasicLTI.xml I_1d0799b4-aa30-342e-a35b-f5d94bde2097_R/BasicLTI.xml I_1d0c3cf7-49d5-3703-9224-d8c84378a494_1_R/BasicLTI.xml I_1d0f46a0-4145-3aa7-9390-2152d2324cc3_1_R/BasicLTI.xml I_1d251665-b0ba-38e5-9dd1-c2018a333388_1_R/BasicLTI.xml I_1d448f3d-def9-360d-8fd9-1059566089fd_R/BasicLTI.xml I_1d69d8fe-f0b4-3a45-8a84-23dda26455ce_1_R/BasicLTI.xml I_1d6a2088-78d5-34c4-9350-5150c90da37d_R/BasicLTI.xml I_1d721f51-39c0-3996-8541-9e6607e03500_1_R/BasicLTI.xml I_1d74a62e-b9aa-3aad-ab3f-b64f3caf1854_R/BasicLTI.xml I_1d92504b-c338-39d8-958f-ab6abc12e4a0_R/BasicLTI.xml I_1da167c2-a023-3358-b97a-3576c869b14f_1_R/BasicLTI.xml I_1da6ee14-925c-35a8-9262-5b2378f022a3_R/BasicLTI.xml I_1dbde513-debf-3cfe-96d0-50c7f9206ff3_R/BasicLTI.xml I_1dc2bec1-9ae6-3a5d-b491-c02aba30586a_1_R/BasicLTI.xml I_1dc5b8b9-f49e-39eb-ae50-2272f1b66294_1_R/BasicLTI.xml I_1dc5b8b9-f49e-39eb-ae50-2272f1b66294_3_R/BasicLTI.xml I_1dc5b8b9-f49e-39eb-ae50-2272f1b66294_5_R/BasicLTI.xml I_1dc5b8b9-f49e-39eb-ae50-2272f1b66294_7_R/BasicLTI.xml I_1dc5b8b9-f49e-39eb-ae50-2272f1b66294_9_R/BasicLTI.xml I_1dcd088d-a574-3edd-a4c5-7a753fea024b_1_R/BasicLTI.xml I_1def39cf-a166-3ae8-acee-33fccbe5dfbd_R/BasicLTI.xml I_1e003521-b37e-38fe-b1e6-e12003ae58a6_1_R/BasicLTI.xml I_1e0803b1-e898-3c0a-9b4a-e38eef7e0e53_1_R/BasicLTI.xml I_1e20b09d-71e6-3515-a9c9-0580152156e7_1_R/BasicLTI.xml I_1e2f7888-bee4-392f-baf7-0b4ab37a16e9_R/BasicLTI.xml I_1e3e2fd9-2457-34f8-960f-10641a82c4ab_R/BasicLTI.xml I_1e440a43-ba11-3dcd-b52a-f8acc26838aa_1_R/BasicLTI.xml I_1e58ab80-1b1e-3074-9f45-20c37d1d24fe_1_R/BasicLTI.xml I_1e593297-8fab-3945-b46b-6846d2798ab0_R/BasicLTI.xml I_1e803bd7-3156-306d-89f9-078844624300_1_R/BasicLTI.xml I_1ea26b84-7c0b-3c23-a76b-3119799236ed_R/BasicLTI.xml I_1ebb761f-2c31-3c08-ae67-f90907cb1e69_1_R/BasicLTI.xml I_1ef8e45d-89eb-35c3-8265-2c9c4ccb1864_1_R/BasicLTI.xml I_1f3f40fe-711a-3bff-be18-cb93ddb970f7_1_R/BasicLTI.xml I_1f41d239-c5f6-3324-bb09-492b0662bb65_1_R/BasicLTI.xml I_1f53a297-a0b2-31b0-bb58-98173adab6a1_1_R/BasicLTI.xml I_1f53a297-a0b2-31b0-bb58-98173adab6a1_3_R/BasicLTI.xml I_1f5b66b2-695d-36b9-b587-fd33a705d385_R/BasicLTI.xml I_1f7971f0-9fb1-39ad-981e-eca0a87cdbf3_R/BasicLTI.xml I_1fa0f93d-5813-37bb-841d-d27916520f37_R/BasicLTI.xml I_1fa9e0e2-fb24-3753-b95f-0e2b30b41f3c_1_R/BasicLTI.xml I_1fac8112-b969-3a6d-90d6-dfe12deb5d6f_R/BasicLTI.xml I_1fc4da80-101b-3538-8d47-8caabcebe551_1_R/BasicLTI.xml I_1fcb14ab-3657-3bd8-9531-0864f61cb529_1_R/BasicLTI.xml I_1fcf0c1e-f370-3178-a650-79eecc7d40b8_1_R/BasicLTI.xml I_1fcf0c1e-f370-3178-a650-79eecc7d40b8_3_R/BasicLTI.xml I_1fcf0c1e-f370-3178-a650-79eecc7d40b8_5_R/BasicLTI.xml I_1fe99c5e-d2b1-3c8b-a128-b9b67ab7cd52_1_R/BasicLTI.xml I_1fedb1cb-fbb1-3f55-9783-30d10c8c6160_1_R/BasicLTI.xml I_1ffe6802-1700-31b2-8a61-dd10b000b4b2_1_R/BasicLTI.xml I_2017dcc3-e626-3fc5-b338-a5125a531f32_1_R/BasicLTI.xml I_2017dcc3-e626-3fc5-b338-a5125a531f32_3_R/BasicLTI.xml I_2017dcc3-e626-3fc5-b338-a5125a531f32_5_R/BasicLTI.xml I_201f6e56-d9e4-3494-8dca-ac9e68cc465d_1_R/BasicLTI.xml I_20275f9a-79c7-39ab-9109-25d3acb1f778_1_R/BasicLTI.xml I_203294f8-bea2-3ee9-b236-f38f7e272129_R/BasicLTI.xml I_2049acb8-a33c-394a-8dc6-866a733c9ee9_R/BasicLTI.xml I_204ae409-9ff0-3def-b4b3-d2d6dd2c0563_1_R/BasicLTI.xml I_204ae409-9ff0-3def-b4b3-d2d6dd2c0563_3_R/BasicLTI.xml I_204ae409-9ff0-3def-b4b3-d2d6dd2c0563_5_R/BasicLTI.xml I_204eea02-ef3b-323e-8849-74736a7b7599_1_R/BasicLTI.xml I_2051409f-ea5c-3eb3-a75e-5f328644c1aa_1_R/BasicLTI.xml I_2051409f-ea5c-3eb3-a75e-5f328644c1aa_3_R/BasicLTI.xml I_20612179-be65-353f-ad63-279f05e7fb1c_1_R/BasicLTI.xml I_2064918f-45a7-3eee-b1f8-9d008427b587_1_R/BasicLTI.xml I_20715aa5-8695-31f2-a185-b62779129d03_1_R/BasicLTI.xml I_207983a3-6f6b-3d09-9f13-14e9f3386931_R/BasicLTI.xml I_209331df-0072-3eeb-88b8-40855a746974_R/BasicLTI.xml I_20949467-3ec2-3844-b193-7cdbbae332c6_R/BasicLTI.xml I_20ad2588-3e4c-3e30-a971-217d23885189_1_R/BasicLTI.xml I_20ad7c0a-5e46-3ea5-8864-9999cbac07d4_1_R/BasicLTI.xml I_20c41f4c-3df8-3894-a558-a8202e8354a9_1_R/BasicLTI.xml I_20c41f4c-3df8-3894-a558-a8202e8354a9_3_R/BasicLTI.xml I_20d91c8a-b604-3c07-aec2-3667e52c564e_1_R/BasicLTI.xml I_20d91c8a-b604-3c07-aec2-3667e52c564e_3_R/BasicLTI.xml I_20d91c8a-b604-3c07-aec2-3667e52c564e_5_R/BasicLTI.xml I_20d91c8a-b604-3c07-aec2-3667e52c564e_7_R/BasicLTI.xml I_20d928b2-6a48-3109-a364-b2102397e799_R/BasicLTI.xml I_2105c358-5c65-362f-a5d7-af1c446a97ab_1_R/BasicLTI.xml I_210fce37-a51d-3e84-b10e-c9fe52c8f800_1_R/BasicLTI.xml I_21261689-a869-38b1-bad2-5e2997be512c_1_R/BasicLTI.xml I_2175f32e-288c-3f2b-b934-67562c35c1ec_R/BasicLTI.xml I_2187e6c5-4f78-3575-8c5b-a3656b2b8ccc_1_R/BasicLTI.xml I_21ac11ae-a183-3c07-9558-83e8561c77de_R/BasicLTI.xml I_21b1c47f-a4e7-3113-81dd-5e5befee8732_R/BasicLTI.xml I_21cf71bb-1b26-3cc4-8876-8d433711a9b0_1_R/BasicLTI.xml I_21e432e6-0c95-345c-8b10-a2ce621df9ce_R/BasicLTI.xml I_21e85df7-86cb-33d1-94ce-25f2c462daf0_1_R/BasicLTI.xml I_21ef62f0-b1d7-39e9-abc0-c367a58d4326_1_R/BasicLTI.xml I_2202282b-3a62-319f-8b44-6b530c1dc84b_1_R/BasicLTI.xml I_2209242a-1160-3702-920c-554fcee9d652_1_R/BasicLTI.xml I_221cd67b-1166-3b7b-b965-b9b3aea9c7da_1_R/BasicLTI.xml I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_1_R/BasicLTI.xml I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_3_R/BasicLTI.xml I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_5_R/BasicLTI.xml I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_7_R/BasicLTI.xml I_2238b5be-1009-38f1-9d8c-50d9524a4a2b_9_R/BasicLTI.xml I_2245f3bd-b799-3a5a-b04b-5c9f7bf7b309_R/BasicLTI.xml I_2246ad6d-e0a5-3203-bb30-de78026dcfab_R/BasicLTI.xml I_227cf9c3-fe03-3525-bcb7-468153f35d8f_1_R/BasicLTI.xml I_227d4d96-995a-35b3-a486-74b442984fdb_R/BasicLTI.xml I_22841b2b-482c-3bda-896c-dbdb775ff1c4_1_R/BasicLTI.xml I_229065b8-b0ab-365a-81e6-de93dec5e706_R/BasicLTI.xml I_22adb4de-16a7-398c-87e1-bf248467bfad_1_R/BasicLTI.xml I_22b63c61-9e28-3c3f-8489-8385c527b152_1_R/BasicLTI.xml I_22cae7e2-9878-3a17-a4b8-d6ad4cfc9ebc_1_R/BasicLTI.xml I_22ddfe20-46c2-3362-95fd-92df06a3c76e_R/BasicLTI.xml I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_1_R/BasicLTI.xml I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_3_R/BasicLTI.xml I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_5_R/BasicLTI.xml I_22e2a68d-11e5-3fc3-911d-0bdaa0b461d1_7_R/BasicLTI.xml I_22e40c57-f40a-39af-a55b-95d1600386f5_1_R/BasicLTI.xml I_22f17c59-e1ad-3366-8a19-25e286e242c3_R/BasicLTI.xml I_23210ac7-1eab-3c8e-99be-6969b3d037e5_R/BasicLTI.xml I_2333b200-d29d-31fb-89db-c769077d68e5_R/BasicLTI.xml I_23491052-0e9d-3337-899d-b7993d7f6638_1_R/BasicLTI.xml I_234d1e37-7879-34ba-9308-2d0c86d84c5b_R/BasicLTI.xml I_236294f9-296d-3142-807f-fd27f5c268c5_1_R/BasicLTI.xml I_236294f9-296d-3142-807f-fd27f5c268c5_3_R/BasicLTI.xml I_236294f9-296d-3142-807f-fd27f5c268c5_5_R/BasicLTI.xml I_236294f9-296d-3142-807f-fd27f5c268c5_7_R/BasicLTI.xml I_236294f9-296d-3142-807f-fd27f5c268c5_9_R/BasicLTI.xml I_23642dea-f69f-3063-b104-07d274a4b4ef_R/BasicLTI.xml I_23682182-2544-3165-8d04-c8f7d4f4b54c_R/BasicLTI.xml I_236d1415-0304-309e-9cd7-e25c1d3841b2_R/BasicLTI.xml I_2379ff91-e48b-3561-91a0-43a20fb159a2_R/BasicLTI.xml I_238f17c7-3809-33d9-8b86-8b7776e98f72_R/BasicLTI.xml I_239a1bf0-c84d-3a7f-9a52-239fa70e11b6_R/BasicLTI.xml I_23ae6868-110c-38f7-b9b3-bda3202ca5e1_1_R/BasicLTI.xml I_23b82881-7f80-33a7-b0b7-7cf01fde23fb_R/BasicLTI.xml I_23d192f4-cd5e-37fc-9150-fb7e7384014b_R/BasicLTI.xml I_23ddc061-3e96-3092-b444-ca1f1158ad5a_1_R/BasicLTI.xml I_23e0a7a3-0c8f-32db-98b1-4faead3e14f5_R/BasicLTI.xml I_23e8fa82-885e-3d52-964a-b9751302accc_R/BasicLTI.xml I_23efec98-528f-303e-874e-40e4f6d13321_1_R/BasicLTI.xml I_241b6aaa-8d28-3d6e-9a63-38a4331a3e89_R/BasicLTI.xml I_241fd91f-04d5-3ec7-815c-57a658c868dc_1_R/BasicLTI.xml I_241fd91f-04d5-3ec7-815c-57a658c868dc_3_R/BasicLTI.xml I_241fd91f-04d5-3ec7-815c-57a658c868dc_5_R/BasicLTI.xml I_24397ca3-a9e8-3b2c-b727-b26b3081ea0b_R/BasicLTI.xml I_24692204-f8d8-31ee-8278-b011d7c013b3_1_R/BasicLTI.xml I_2482525d-1283-3f7d-88ae-c6bab0526852_1_R/BasicLTI.xml I_2496f49e-3938-350c-b6b5-068c280236d7_R/BasicLTI.xml I_249c045c-ab6b-38d6-b19c-96ed315e2c96_1_R/BasicLTI.xml I_249c045c-ab6b-38d6-b19c-96ed315e2c96_3_R/BasicLTI.xml I_24bf5af8-2f8a-323b-9848-ef9b42050386_1_R/BasicLTI.xml I_24cc3e49-a2a0-3cc4-ae1c-b0ca40f997ba_1_R/BasicLTI.xml I_24ee765a-46b6-3435-b288-e5b49bd92c20_1_R/BasicLTI.xml I_24ee765a-46b6-3435-b288-e5b49bd92c20_3_R/BasicLTI.xml I_24ee765a-46b6-3435-b288-e5b49bd92c20_5_R/BasicLTI.xml I_24ee765a-46b6-3435-b288-e5b49bd92c20_7_R/BasicLTI.xml I_24f820c4-7661-396a-9cf8-759c6e9d2fcb_1_R/BasicLTI.xml I_2507bf19-68f7-3c8d-a27a-97f6ff76a3b1_R/BasicLTI.xml I_252008bd-b400-3c37-ab08-00f6a0595e4d_1_R/BasicLTI.xml I_2525cc39-92c8-3fbb-8070-f4ae06811208_1_R/BasicLTI.xml I_252ab9b3-bc82-3be6-a5a9-c41e13455dce_1_R/BasicLTI.xml I_254a663c-4562-3a3a-b50f-ffc1193c0a83_1_R/BasicLTI.xml I_254b5afd-3111-39e5-b502-57aa53acf581_R/BasicLTI.xml I_2586ec3e-09a7-3ce6-8b36-f89d7f5ac0cf_1_R/BasicLTI.xml I_2586f5e5-7e2f-380f-8faf-38970349b299_1_R/BasicLTI.xml I_2586f5e5-7e2f-380f-8faf-38970349b299_3_R/BasicLTI.xml I_2586f5e5-7e2f-380f-8faf-38970349b299_5_R/BasicLTI.xml I_2586f5e5-7e2f-380f-8faf-38970349b299_7_R/BasicLTI.xml I_25a0113e-5e1e-392d-81df-ee289e17e878_R/BasicLTI.xml I_25a3903e-568c-319a-a5a1-ce6f2d883717_1_R/BasicLTI.xml I_25ac3b3f-caec-360a-98b1-672ba00dce31_1_R/BasicLTI.xml I_25e0f91d-6d01-33ba-8a62-fe8716fbb7c7_1_R/BasicLTI.xml I_25ee9142-c712-3c7e-b250-05992b98afcd_R/BasicLTI.xml I_2603b589-0235-3c10-bce4-0ba6bffc0a53_1_R/BasicLTI.xml I_26078341-51fe-32bf-bd29-66d05b862ae0_R/BasicLTI.xml I_261e49ea-2429-358c-bdce-8aed5511a2e2_1_R/BasicLTI.xml I_26221b16-0e9c-3389-8e58-89472e334043_1_R/BasicLTI.xml I_26221b16-0e9c-3389-8e58-89472e334043_3_R/BasicLTI.xml I_26221b16-0e9c-3389-8e58-89472e334043_5_R/BasicLTI.xml I_26221b16-0e9c-3389-8e58-89472e334043_7_R/BasicLTI.xml I_262a7c13-1d7f-3c25-aa17-6cec4c952307_1_R/BasicLTI.xml I_262ef172-bd6b-3886-b5e1-a7a7855fb39b_1_R/BasicLTI.xml I_263b4cf5-7485-3105-aa8b-52e65494a7cc_1_R/BasicLTI.xml I_26473a41-c0a6-3b98-a4d4-edca5bb01bd4_R/BasicLTI.xml I_265220f9-2b2d-30d1-bdf4-5912a910c529_1_R/BasicLTI.xml I_265220f9-2b2d-30d1-bdf4-5912a910c529_3_R/BasicLTI.xml I_26709bbe-e728-307a-876d-305c9813c5a7_1_R/BasicLTI.xml I_26b53302-b1f2-30cc-b3c9-743d19720268_1_R/BasicLTI.xml I_26b53302-b1f2-30cc-b3c9-743d19720268_3_R/BasicLTI.xml I_26b53302-b1f2-30cc-b3c9-743d19720268_5_R/BasicLTI.xml I_26bc5089-0feb-3fba-bab1-dbc61fe1d161_1_R/BasicLTI.xml I_26bc5089-0feb-3fba-bab1-dbc61fe1d161_3_R/BasicLTI.xml I_26bc5089-0feb-3fba-bab1-dbc61fe1d161_5_R/BasicLTI.xml I_26bc5089-0feb-3fba-bab1-dbc61fe1d161_7_R/BasicLTI.xml I_2718ff91-74dd-304a-bbae-485121c941fb_1_R/BasicLTI.xml I_2737b2f7-7d4a-3819-b157-d3eac53342e4_1_R/BasicLTI.xml I_2737b2f7-7d4a-3819-b157-d3eac53342e4_3_R/BasicLTI.xml I_2737b2f7-7d4a-3819-b157-d3eac53342e4_5_R/BasicLTI.xml I_2737b2f7-7d4a-3819-b157-d3eac53342e4_7_R/BasicLTI.xml I_273d7b58-33fc-35ce-922e-e0cb23977f13_R/BasicLTI.xml I_27424347-614c-3cda-8b8f-46407952e544_1_R/BasicLTI.xml I_27424347-614c-3cda-8b8f-46407952e544_3_R/BasicLTI.xml I_274e85c9-841a-3fad-a7bc-bcaab1e1e5de_1_R/BasicLTI.xml I_2783835b-95ba-37a8-bccd-a864536e0a40_R/BasicLTI.xml I_27a0b3b7-4a13-3bbf-b008-538dfe21192b_1_R/BasicLTI.xml I_27b7733a-3e36-3fdb-b1e5-1deb8d588380_R/BasicLTI.xml I_27bcc5da-3e8b-3e3f-b19e-c33d1b6e4c53_1_R/BasicLTI.xml I_27c6e205-686a-3a2e-8a6b-4eecace5b796_R/BasicLTI.xml I_27e154ca-d62c-35e7-a11c-a85255a3dffa_1_R/BasicLTI.xml I_27eae7c9-3a5f-3c16-a4fe-67ec7d0ac844_1_R/BasicLTI.xml I_27eae7c9-3a5f-3c16-a4fe-67ec7d0ac844_3_R/BasicLTI.xml I_27ee2e80-7b3d-3273-92c6-bc818f522ca5_R/BasicLTI.xml I_27ff494d-11f7-3d29-8181-82b127bf89f6_R/BasicLTI.xml I_28207ddd-73b9-3153-b8c5-5b0d363670ff_1_R/BasicLTI.xml I_283a1c2c-bd91-330a-a031-648a4a2e2013_1_R/BasicLTI.xml I_285f4781-59aa-35f7-9541-37b4d5433853_1_R/BasicLTI.xml I_286472ac-676e-3789-813a-9e01fb303ac8_1_R/BasicLTI.xml I_287d034c-e934-3c79-927b-61f2e36e10b0_1_R/BasicLTI.xml I_289169c8-cfb3-3239-8d70-2fac5c21a3fb_1_R/BasicLTI.xml I_289169c8-cfb3-3239-8d70-2fac5c21a3fb_3_R/BasicLTI.xml I_28a3d56e-4b98-352c-8e26-6cdd087f4cea_R/BasicLTI.xml I_28aa42cf-cf04-34df-ba7e-5e65becc70dd_1_R/BasicLTI.xml I_28aa42cf-cf04-34df-ba7e-5e65becc70dd_3_R/BasicLTI.xml I_28aa42cf-cf04-34df-ba7e-5e65becc70dd_5_R/BasicLTI.xml I_28aa42cf-cf04-34df-ba7e-5e65becc70dd_7_R/BasicLTI.xml I_28aa42cf-cf04-34df-ba7e-5e65becc70dd_9_R/BasicLTI.xml I_28ad19e4-502c-347c-8a78-7efc3e49a526_1_R/BasicLTI.xml I_28e3f73e-91d0-347d-99ff-6dc55fba9b66_1_R/BasicLTI.xml I_28e3f73e-91d0-347d-99ff-6dc55fba9b66_3_R/BasicLTI.xml I_28e44c3f-2d82-3810-b95c-1ce0b40ba61c_1_R/BasicLTI.xml I_28e44c3f-2d82-3810-b95c-1ce0b40ba61c_3_R/BasicLTI.xml I_28e44c3f-2d82-3810-b95c-1ce0b40ba61c_5_R/BasicLTI.xml I_28e44c3f-2d82-3810-b95c-1ce0b40ba61c_7_R/BasicLTI.xml I_28e44c3f-2d82-3810-b95c-1ce0b40ba61c_9_R/BasicLTI.xml I_290ea16e-1e36-3a4e-a8d3-dc105f8a66d1_R/BasicLTI.xml I_292deb4e-8e58-34e7-aca1-bc3cf384493f_1_R/BasicLTI.xml I_293215e0-3eac-33b6-b95b-a058fcabca08_R/BasicLTI.xml I_293b316b-800e-3d88-90c8-f5188b4b9776_1_R/BasicLTI.xml I_293ea443-6486-3d76-bb8f-f1e630a4c093_R/BasicLTI.xml I_2949a6cd-cf67-3b25-b093-38c5f755d876_1_R/BasicLTI.xml I_29515368-85d5-3656-92d0-653fb5db03ae_1_R/BasicLTI.xml I_2958a6f3-2f90-3878-ac49-40827be688b2_1_R/BasicLTI.xml I_295ba321-8676-3744-897e-a460ce2e9b70_R/BasicLTI.xml I_2964bda4-2932-3c57-8dbd-e23aadff77ee_1_R/BasicLTI.xml I_296c7842-0945-3726-b753-d8bccc47bc49_R/BasicLTI.xml I_29709643-3087-3ca6-9246-94df55b699b6_R/BasicLTI.xml I_2983a8ed-7a5c-3566-a50f-7dbd5a3de6d1_1_R/BasicLTI.xml I_29a48801-862e-3925-a36c-65d9e101bc73_1_R/BasicLTI.xml I_29b4bdea-ecae-375c-b2f1-f9830a0c85bd_R/BasicLTI.xml I_29ba23f9-64f5-3334-a5e4-bb0747a2e719_1_R/BasicLTI.xml I_29c5a431-7eca-3253-97fb-e0d0aca04bec_R/BasicLTI.xml I_29da0d83-c2ec-36b6-8a7e-1b8048ca47a2_1_R/BasicLTI.xml I_2a0b112e-02fc-3727-99cb-5609deb6c11d_1_R/BasicLTI.xml I_2a146582-ea28-39d3-a098-b9884285bf4d_1_R/BasicLTI.xml I_2a146582-ea28-39d3-a098-b9884285bf4d_3_R/BasicLTI.xml I_2a146582-ea28-39d3-a098-b9884285bf4d_5_R/BasicLTI.xml I_2a146582-ea28-39d3-a098-b9884285bf4d_7_R/BasicLTI.xml I_2a178120-87ac-3dc2-8c53-dff41c52c267_1_R/BasicLTI.xml I_2a2a10d6-9d17-3836-a400-58014f4be891_1_R/BasicLTI.xml I_2a315211-dbf8-3ac0-a1be-99bd148a9114_1_R/BasicLTI.xml I_2a398412-232f-384a-aba1-86e235c39886_1_R/BasicLTI.xml I_2a398412-232f-384a-aba1-86e235c39886_3_R/BasicLTI.xml I_2a398412-232f-384a-aba1-86e235c39886_5_R/BasicLTI.xml I_2a398412-232f-384a-aba1-86e235c39886_7_R/BasicLTI.xml I_2a398412-232f-384a-aba1-86e235c39886_9_R/BasicLTI.xml I_2a4a2385-e2f2-3b3c-a8fa-8acae7f3052e_1_R/BasicLTI.xml I_2a672adc-682e-3ba2-9950-b3285baac0ae_1_R/BasicLTI.xml I_2a6d2bb3-5862-30b3-8eb5-505d2caa51be_1_R/BasicLTI.xml I_2a7cc298-fb27-357b-8375-9eaf8f1aa003_R/BasicLTI.xml I_2a908c3e-9dce-36f1-afb0-3c7653cd1310_1_R/BasicLTI.xml I_2a9b90a6-0b75-34a8-8126-d9e1c37fd4b5_R/BasicLTI.xml I_2ac354d7-0ccc-3b73-9672-939107c0047b_1_R/BasicLTI.xml I_2ac354d7-0ccc-3b73-9672-939107c0047b_3_R/BasicLTI.xml I_2ad5bf09-75f8-3e88-925c-4a37809a3f6a_1_R/BasicLTI.xml I_2ad9bdf9-5226-33d3-86f9-322798d3371e_1_R/BasicLTI.xml I_2ad9bdf9-5226-33d3-86f9-322798d3371e_3_R/BasicLTI.xml I_2ae0daf8-80a5-354c-9524-8f1f3ffc1c74_R/BasicLTI.xml I_2af46e93-1230-3e9a-b349-e69c82a7d743_R/BasicLTI.xml I_2b021e81-44c3-3693-afe5-a2729ce21fc6_R/BasicLTI.xml I_2b0bd427-a612-3423-a20c-04a10381eeee_R/BasicLTI.xml I_2b1eb2bb-166d-3242-ae00-35dc7c916aad_1_R/BasicLTI.xml I_2b2d644f-ed58-350b-9daf-ed43ccc21606_1_R/BasicLTI.xml I_2b30f315-c115-3a6e-b85c-686387ab91bb_1_R/BasicLTI.xml I_2b30f315-c115-3a6e-b85c-686387ab91bb_3_R/BasicLTI.xml I_2b377c10-3ea3-3811-ba2e-6b2d668cd125_1_R/BasicLTI.xml I_2b395cb0-b160-3a58-bd83-eed5cd2779db_1_R/BasicLTI.xml I_2b395cb0-b160-3a58-bd83-eed5cd2779db_3_R/BasicLTI.xml I_2b45c884-2c54-39c8-aa4d-2869f4108b78_1_R/BasicLTI.xml I_2b4a8321-bb5e-399f-b98f-7f9ec4968705_1_R/BasicLTI.xml I_2b4a8321-bb5e-399f-b98f-7f9ec4968705_3_R/BasicLTI.xml I_2b55541e-6df6-322b-9c87-10c7cf4d1626_1_R/BasicLTI.xml I_2b6e9d06-2702-3f65-a2c2-5e12ea6f36b0_1_R/BasicLTI.xml I_2b80bbbf-69bf-3cee-bd33-8e2814372a28_1_R/BasicLTI.xml I_2ba262c5-0fcd-390a-b7d0-805babb3e312_R/BasicLTI.xml I_2bc6b00b-6a02-3f2a-a5de-f9c4e623e033_1_R/BasicLTI.xml I_2bd61ef4-6d63-3122-bf18-fac840eec1bd_1_R/BasicLTI.xml I_2bd61ef4-6d63-3122-bf18-fac840eec1bd_3_R/BasicLTI.xml I_2be10aee-05a8-3e5d-be3f-8a1e70be856c_1_R/BasicLTI.xml I_2bebed0c-417e-3ee9-9ae4-642cda2d062d_1_R/BasicLTI.xml I_2bebed0c-417e-3ee9-9ae4-642cda2d062d_3_R/BasicLTI.xml I_2c38741e-12a2-3930-89f4-53b5c037cdd7_R/BasicLTI.xml I_2c47db5b-7a3b-3cb6-ae1f-ef054dd8b209_1_R/BasicLTI.xml I_2c605436-84bc-346d-9fbc-2fef243f75c6_1_R/BasicLTI.xml I_2c755c9d-ff47-3a47-81c3-b57857830945_R/BasicLTI.xml I_2cc8f722-fe68-3bf3-a3b8-5b8861bd2fec_1_R/BasicLTI.xml I_2cd64aca-5339-36fd-9283-e340b5c725ec_1_R/BasicLTI.xml I_2cd64aca-5339-36fd-9283-e340b5c725ec_3_R/BasicLTI.xml I_2cd64aca-5339-36fd-9283-e340b5c725ec_5_R/BasicLTI.xml I_2cdb12fb-189e-31df-b363-18c79ffbffe1_1_R/BasicLTI.xml I_2ce435f3-7e60-3722-994e-73445eba0bd4_1_R/BasicLTI.xml I_2d021435-cd1d-3c0e-96aa-1b20f1e8b2d1_1_R/BasicLTI.xml I_2d3a264f-9c3b-3ac4-9ebb-163b58b39ff2_1_R/BasicLTI.xml I_2d3bdcb6-00d0-38b6-8251-52a3e5cfc1cd_1_R/BasicLTI.xml I_2d527f36-f8b0-3052-8c70-3537465a4dd0_1_R/BasicLTI.xml I_2d59c880-f27a-30c8-a959-032ffdca5827_1_R/BasicLTI.xml I_2d59edf4-ba7b-3eea-81b7-79f6a51a04b9_1_R/BasicLTI.xml I_2d621e77-2468-30d8-afcf-65dba74bfb34_1_R/BasicLTI.xml I_2d757b0f-a274-3918-a028-3a18a5cd0b75_1_R/BasicLTI.xml I_2d7c619e-1968-37f8-b7b8-19c2c0f348c7_1_R/BasicLTI.xml I_2d8d7576-d718-3937-81a8-b356b8f8a56a_1_R/BasicLTI.xml I_2d8d7576-d718-3937-81a8-b356b8f8a56a_3_R/BasicLTI.xml I_2d8d7576-d718-3937-81a8-b356b8f8a56a_5_R/BasicLTI.xml I_2d8d7576-d718-3937-81a8-b356b8f8a56a_7_R/BasicLTI.xml I_2d973ead-fec2-3238-a15f-a2f669598dad_R/BasicLTI.xml I_2da39503-de9b-3cb6-9fe8-3dc917891633_R/BasicLTI.xml I_2dd17ab2-9def-3eab-b2c5-5b0aea1084a7_R/BasicLTI.xml I_2dfacad0-63c9-319c-bc15-fe573d1f12d0_R/BasicLTI.xml I_2e3e8387-319d-3840-87ab-439675a443e9_1_R/BasicLTI.xml I_2e5ed880-04e8-30aa-b446-35e3b8615e46_1_R/BasicLTI.xml I_2e945055-2052-357b-b92a-10ab0df84129_1_R/BasicLTI.xml I_2e945055-2052-357b-b92a-10ab0df84129_3_R/BasicLTI.xml I_2e945055-2052-357b-b92a-10ab0df84129_5_R/BasicLTI.xml I_2e945055-2052-357b-b92a-10ab0df84129_7_R/BasicLTI.xml I_2e9811e1-5721-3417-9b27-ff2c53bb74f5_1_R/BasicLTI.xml I_2ea178c5-3ae0-3a0d-980e-d9c7887edb1c_R/BasicLTI.xml I_2ea6a401-20f6-3b92-ac3f-332e7a552b9f_1_R/BasicLTI.xml I_2eb959c8-d9e5-3017-a256-a7cd79e9d5bf_1_R/BasicLTI.xml I_2ebf1097-fd86-3575-8578-524fc2990e25_R/BasicLTI.xml I_2ec99c87-92d2-3632-8211-2dbdab381294_1_R/BasicLTI.xml I_2ef631f7-d7af-3efb-b34f-d9695192a95d_1_R/BasicLTI.xml I_2f09bb91-6598-3869-b8ad-e41ad1e2ec68_1_R/BasicLTI.xml I_2f20babb-7efc-3d87-b7f9-bd4021596e4f_1_R/BasicLTI.xml I_2f259e2b-2341-3ccf-9840-25e6f143cabf_1_R/BasicLTI.xml I_2f259e2b-2341-3ccf-9840-25e6f143cabf_3_R/BasicLTI.xml I_2f269d0c-020a-3f29-aed0-d897ce343298_1_R/BasicLTI.xml I_2f3bffe9-13b2-30d6-82a4-af0648295d99_1_R/BasicLTI.xml I_2f432390-dac6-3fe4-b7ff-01effe388b0a_R/BasicLTI.xml I_2f4496c3-ad4d-315d-9499-3346f3d85b10_1_R/BasicLTI.xml I_2f4fe02b-ae95-3500-8875-9498f3626f13_1_R/BasicLTI.xml I_2f61921e-4112-3467-977c-2f51ac58f23c_1_R/BasicLTI.xml I_2f8931cd-3522-33b5-8e8d-b937b682791f_1_R/BasicLTI.xml I_2f99835d-f9bb-3160-814d-ece977f7a7eb_1_R/BasicLTI.xml I_2fcc7be1-f3b2-37be-b4da-895b29d569e7_1_R/BasicLTI.xml I_2fe9686a-6d4e-37ce-8d1c-a8038e5ea318_1_R/BasicLTI.xml I_2fe9686a-6d4e-37ce-8d1c-a8038e5ea318_3_R/BasicLTI.xml I_30300d48-94a1-3e29-a0f3-d44025f5c2e1_1_R/BasicLTI.xml I_3031fbf6-c346-36a0-8ea8-21321d570cdf_1_R/BasicLTI.xml I_3031fbf6-c346-36a0-8ea8-21321d570cdf_3_R/BasicLTI.xml I_3031fbf6-c346-36a0-8ea8-21321d570cdf_5_R/BasicLTI.xml I_3031fbf6-c346-36a0-8ea8-21321d570cdf_7_R/BasicLTI.xml I_3035fee8-5f04-3055-9148-688e0c57e85f_R/BasicLTI.xml I_304a3b92-cecc-3673-a555-88f9372e8034_R/BasicLTI.xml I_307645ed-790d-3e27-85db-174f5baaf937_1_R/BasicLTI.xml I_307be909-4aa8-3f13-9a14-6b024ce72579_1_R/BasicLTI.xml I_308ae63f-3136-3d83-9ab4-0857d2d14888_1_R/BasicLTI.xml I_30abdc13-fecd-3c7a-9617-f9e1cd7d8599_1_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_11_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_1_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_3_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_5_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_7_R/BasicLTI.xml I_30fd4861-ca37-3d33-9f1c-711f3ae8f489_9_R/BasicLTI.xml I_31015808-84df-3827-92ab-e3994b275b16_1_R/BasicLTI.xml I_31037d8c-cd59-31f2-8acb-869bd1cacee8_1_R/BasicLTI.xml I_3106eb72-238a-3f07-b093-f82dc6a7174d_1_R/BasicLTI.xml I_312158aa-9121-3d06-bdb8-81ff53b270ee_R/BasicLTI.xml I_31510cea-ba91-319b-b5f1-1c10f4489a49_R/BasicLTI.xml I_3151c7ec-cc53-3fcf-80c2-ce6959c2d565_R/BasicLTI.xml I_31542840-78cb-3c86-ab84-e637806632f4_1_R/BasicLTI.xml I_3155722e-d1bd-3c18-b9d3-d9549f7d3462_1_R/BasicLTI.xml I_31a10454-e1b2-390d-bf4a-327cea6a3bee_1_R/BasicLTI.xml I_31cbf584-6859-3ab6-ae05-571f99c8dd32_1_R/BasicLTI.xml I_31cde6ae-0043-38d6-863e-064100522155_R/BasicLTI.xml I_31fafefe-8409-3bb2-a592-ce883b173580_1_R/BasicLTI.xml I_3226f352-04f5-349a-9d63-d3df99457f18_R/BasicLTI.xml I_324454b1-3848-39c7-87e1-eaca71be0e89_1_R/BasicLTI.xml I_324454b1-3848-39c7-87e1-eaca71be0e89_3_R/BasicLTI.xml I_324912fd-c5b7-3f02-b86e-7617bc8911bf_1_R/BasicLTI.xml I_3257c3ed-eef5-3a70-917d-1ae71135456b_1_R/BasicLTI.xml I_327ebf53-1fe3-35d4-82bd-cbc209a1ae49_1_R/BasicLTI.xml I_32c0ee1e-3166-3469-9fa5-3444cfedf841_1_R/BasicLTI.xml I_32c97d13-cb1e-30fb-b1a0-cfe4baa7076d_1_R/BasicLTI.xml I_32c97d13-cb1e-30fb-b1a0-cfe4baa7076d_3_R/BasicLTI.xml I_32c97d13-cb1e-30fb-b1a0-cfe4baa7076d_5_R/BasicLTI.xml I_32c97d13-cb1e-30fb-b1a0-cfe4baa7076d_7_R/BasicLTI.xml I_32cfa431-fc2c-3d89-9a96-96a42597a339_R/BasicLTI.xml I_32e313ac-cc50-3d65-99dd-0a0093328c33_R/BasicLTI.xml I_3309637a-c77a-3248-a3f4-b88970630e65_1_R/BasicLTI.xml I_3309637a-c77a-3248-a3f4-b88970630e65_3_R/BasicLTI.xml I_3309637a-c77a-3248-a3f4-b88970630e65_5_R/BasicLTI.xml I_33116483-0c2f-39a2-a927-fb2a73b86c2f_1_R/BasicLTI.xml I_332d792a-a87a-3060-8919-d58e4a93d3a1_1_R/BasicLTI.xml I_3336a444-e21d-3e82-97dc-badc22f78573_1_R/BasicLTI.xml I_3338c86b-7c2c-3774-8944-16e86419d90f_1_R/BasicLTI.xml I_337335c0-e329-3363-a596-0d601287e21e_1_R/BasicLTI.xml I_3379d96d-17e5-3b69-b70d-e87e94196923_1_R/BasicLTI.xml I_3379d96d-17e5-3b69-b70d-e87e94196923_3_R/BasicLTI.xml I_339635e7-255c-32eb-a9e5-2fe306a2b04c_R/BasicLTI.xml I_33a7f7cc-8149-34e3-9ebc-701518e6231e_1_R/BasicLTI.xml I_33b0aa16-b987-3323-b920-d3665fdaec0f_1_R/BasicLTI.xml I_33b7582a-e1a5-3cc5-aab9-63640fe910e6_1_R/BasicLTI.xml I_33c69595-b8ae-3c6f-9dc2-347b2e51570a_1_R/BasicLTI.xml I_33db77e9-a9b3-3d34-bfb3-38a891662592_1_R/BasicLTI.xml I_33eb9f1c-6620-3a1b-977a-1b58c8e80437_R/BasicLTI.xml I_33ebcc30-3401-37ca-80ed-3692daf4c8ee_R/BasicLTI.xml I_33fd44c7-53af-32ef-8bfe-012d2d21120f_R/BasicLTI.xml I_3403d4ac-5992-3dbe-a017-70d5a8646dff_1_R/BasicLTI.xml I_3415ca28-cd7e-33e5-bcb5-e2f18aae58d5_1_R/BasicLTI.xml I_341862b3-7116-3305-bf8d-319355a3e330_1_R/BasicLTI.xml I_341862b3-7116-3305-bf8d-319355a3e330_3_R/BasicLTI.xml I_341862b3-7116-3305-bf8d-319355a3e330_5_R/BasicLTI.xml I_341862b3-7116-3305-bf8d-319355a3e330_7_R/BasicLTI.xml I_341970c1-8ccf-356a-a8a9-7ee970fb42e8_1_R/BasicLTI.xml I_341970c1-8ccf-356a-a8a9-7ee970fb42e8_3_R/BasicLTI.xml I_343a1831-25d4-3a97-a600-89513c91e5a5_1_R/BasicLTI.xml I_343fe021-b854-3624-a257-db4ead5e17e9_1_R/BasicLTI.xml I_343fe021-b854-3624-a257-db4ead5e17e9_3_R/BasicLTI.xml I_343fe021-b854-3624-a257-db4ead5e17e9_5_R/BasicLTI.xml I_343fe021-b854-3624-a257-db4ead5e17e9_7_R/BasicLTI.xml I_344df115-7168-3219-89b2-4a15ef4e1a6a_R/BasicLTI.xml I_344ec4ea-3153-34a6-8d20-f72415411e86_1_R/BasicLTI.xml I_344ec4ea-3153-34a6-8d20-f72415411e86_3_R/BasicLTI.xml I_345352ab-6db6-308c-b1f1-ffa3be3d73bd_R/BasicLTI.xml I_345fa2d3-1a3a-3a92-8bf7-db97f5cb7a32_R/BasicLTI.xml I_346179c3-b2de-39b0-b104-8a64ded02f3c_1_R/BasicLTI.xml I_346179c3-b2de-39b0-b104-8a64ded02f3c_3_R/BasicLTI.xml I_34721174-4abf-3077-86f2-2e486ab02a40_1_R/BasicLTI.xml I_34a4f9f7-8e65-36f5-9110-9ce844367faa_1_R/BasicLTI.xml I_34e4cdec-84d3-35dc-9191-3276f0d47f78_1_R/BasicLTI.xml I_34e4cdec-84d3-35dc-9191-3276f0d47f78_3_R/BasicLTI.xml I_34e4cdec-84d3-35dc-9191-3276f0d47f78_5_R/BasicLTI.xml I_34e4cdec-84d3-35dc-9191-3276f0d47f78_7_R/BasicLTI.xml I_34e61c74-728f-3a58-a093-3ac9ae3deb81_R/BasicLTI.xml I_34f49f35-e8aa-3633-abdd-46f882384727_1_R/BasicLTI.xml I_34fb259f-4982-31cf-bcf3-b28e1c33febb_R/BasicLTI.xml I_352823a2-0e8b-3f10-8dd6-cc18ee0b42d1_1_R/BasicLTI.xml I_352823a2-0e8b-3f10-8dd6-cc18ee0b42d1_3_R/BasicLTI.xml I_35619e1d-00d5-3ea3-a3cf-a40c532b5673_1_R/BasicLTI.xml I_358c66f8-2cdd-39f7-ba6a-d8e165a3f45e_1_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_11_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_1_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_3_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_5_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_7_R/BasicLTI.xml I_359214fd-00c7-3acc-ba9b-2c111449d7bd_9_R/BasicLTI.xml I_35a00b90-ae8c-3b67-8373-ad644b9e119e_1_R/BasicLTI.xml I_35ac8716-2ca4-3fff-a494-2e2bb29aea8a_1_R/BasicLTI.xml I_35cbb404-abc5-38fc-a91a-8ebc9d22becf_1_R/BasicLTI.xml I_35d051bb-2a31-37f5-bdda-164d208c1870_1_R/BasicLTI.xml I_35e2b974-b728-37d0-a2b2-0cea4e4ca1f2_1_R/BasicLTI.xml I_35e5c819-b91f-38a4-8b04-6da31e2c1b76_1_R/BasicLTI.xml I_36015afc-498b-3883-a298-765e6d78092d_1_R/BasicLTI.xml I_3608f5aa-bd8b-3ea0-938d-fa925a10ae52_R/BasicLTI.xml I_360a2340-e335-3e26-afa3-01cb65827bfa_R/BasicLTI.xml I_36156937-9718-3f67-a8c9-0c7335ada278_R/BasicLTI.xml I_3623b1b5-0a1c-3ce8-8062-2044fe71b59c_1_R/BasicLTI.xml I_3623b1b5-0a1c-3ce8-8062-2044fe71b59c_3_R/BasicLTI.xml I_3623b1b5-0a1c-3ce8-8062-2044fe71b59c_5_R/BasicLTI.xml I_3623b1b5-0a1c-3ce8-8062-2044fe71b59c_7_R/BasicLTI.xml I_36500a2e-8c78-3c11-828a-c08fd87333c3_1_R/BasicLTI.xml I_36500a2e-8c78-3c11-828a-c08fd87333c3_3_R/BasicLTI.xml I_3653d47e-0e75-38c5-9ebc-e328e8525c18_1_R/BasicLTI.xml I_365e995c-f25f-3bfe-8362-221a3db5565b_1_R/BasicLTI.xml I_365f6c50-072f-30d4-a9ca-e018c951bbaf_R/BasicLTI.xml I_36861b9f-4396-3563-9e14-618d4cfbff55_R/BasicLTI.xml I_368ffce2-a129-3f58-94b0-ee35ad0f011c_R/BasicLTI.xml I_36be5676-541a-3522-aea4-84faab5806d7_1_R/BasicLTI.xml I_36edbfb7-07f0-342d-92ca-bf62a830dd9c_1_R/BasicLTI.xml I_36ff1a05-8c57-3b0f-96f8-e1ddada9412f_1_R/BasicLTI.xml I_37033b95-8a5b-3f98-81c8-b4c5e7287434_R/BasicLTI.xml I_370668e5-95a6-3d15-a89e-1dfb785aca34_1_R/BasicLTI.xml I_370bacb4-bf55-3ea0-849f-3dc78c004057_R/BasicLTI.xml I_37258c56-11e6-3de4-9327-53d2548130a0_R/BasicLTI.xml I_37280b23-cf29-3684-8f1b-3a1a2d14658b_1_R/BasicLTI.xml I_372e8280-ae24-391b-8ce3-485b2ef7cb2b_R/BasicLTI.xml I_37387fd2-5aa9-3af2-ba27-27ab17062184_1_R/BasicLTI.xml I_37417f06-7639-3ce7-b93e-811d5f1a8911_R/BasicLTI.xml I_374a947e-ea1d-314a-95dc-31f76f80f7ba_1_R/BasicLTI.xml I_374a947e-ea1d-314a-95dc-31f76f80f7ba_3_R/BasicLTI.xml I_374a947e-ea1d-314a-95dc-31f76f80f7ba_5_R/BasicLTI.xml I_374a947e-ea1d-314a-95dc-31f76f80f7ba_7_R/BasicLTI.xml I_37690442-9945-390c-b70c-4cf0843a3f95_R/BasicLTI.xml I_376d047a-095d-38e9-ac22-99cd0b2b742b_1_R/BasicLTI.xml I_376d047a-095d-38e9-ac22-99cd0b2b742b_3_R/BasicLTI.xml I_3773df56-c4ab-35ac-b2dd-4cf6acfb73ee_1_R/BasicLTI.xml I_37b9d57e-92f8-30fb-a66d-852b4eecc0af_R/BasicLTI.xml I_37e56553-9368-3c40-952e-2511d7774c7c_1_R/BasicLTI.xml I_37eb0cf0-8c72-3ce0-90fa-190e952a4bdf_1_R/BasicLTI.xml I_37eb0cf0-8c72-3ce0-90fa-190e952a4bdf_3_R/BasicLTI.xml I_37f003f2-5e0d-32d3-a776-bc1d289649b1_R/BasicLTI.xml I_37f044e8-2ae1-35a4-a22e-1bca91ac05af_1_R/BasicLTI.xml I_37f05b48-133f-3c14-9b3c-cb31c781aa4a_1_R/BasicLTI.xml I_37f05b48-133f-3c14-9b3c-cb31c781aa4a_3_R/BasicLTI.xml I_37f8d48d-16d9-312e-b634-0dc5bb730bbe_1_R/BasicLTI.xml I_38088ea2-9428-3760-bd8f-274379d1e83f_R/BasicLTI.xml I_380ec176-13ca-3df3-9619-9a48fb8e1bae_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_11_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_13_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_1_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_3_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_5_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_7_R/BasicLTI.xml I_381cf5f7-ad01-37f9-a64a-2d836fcd8add_9_R/BasicLTI.xml I_3828cd5d-f2e0-364f-960b-e12eb1b80f17_R/BasicLTI.xml I_384b20be-9ef7-3f5b-adc4-ed8c1ef6b04a_1_R/BasicLTI.xml I_38937a1c-10a5-37b7-8073-c43348e99b19_1_R/BasicLTI.xml I_38a23d46-2e0a-3df0-a1be-7ca74ee8964a_R/BasicLTI.xml I_38b38292-9524-3e1f-b048-4e1455263b3a_1_R/BasicLTI.xml I_38c60e14-de00-3994-bcee-2601a5fd1f34_1_R/BasicLTI.xml I_38c60e14-de00-3994-bcee-2601a5fd1f34_3_R/BasicLTI.xml I_38c60e14-de00-3994-bcee-2601a5fd1f34_5_R/BasicLTI.xml I_38c60e14-de00-3994-bcee-2601a5fd1f34_7_R/BasicLTI.xml I_38c60e14-de00-3994-bcee-2601a5fd1f34_9_R/BasicLTI.xml I_38d96e12-3834-39e9-b8cc-bc2336041749_R/BasicLTI.xml I_38daa741-e6ad-3e65-b70d-4a6ec19237aa_R/BasicLTI.xml I_38e39a19-6366-367b-aba3-02d941b44c20_1_R/BasicLTI.xml I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_1_R/BasicLTI.xml I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_3_R/BasicLTI.xml I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_5_R/BasicLTI.xml I_38ef5fcd-1f7a-359c-9a7a-c5399f8f3e09_7_R/BasicLTI.xml I_38f0331f-cb78-3855-86a7-9f075e145dc0_R/BasicLTI.xml I_38f94e52-8ced-30c6-b4f5-48b2799a62fb_R/BasicLTI.xml I_3927fb71-715b-3804-9080-9d09f52a4691_1_R/BasicLTI.xml I_39293742-176e-3843-9122-6da7f47aab2b_R/BasicLTI.xml I_392d4fce-84ae-341e-8d2b-7777515b1d98_R/BasicLTI.xml I_392e1a2a-16fc-3e39-a2c2-c504e0f8ca29_1_R/BasicLTI.xml I_3995af9f-819b-3359-a0df-58380a566eb5_1_R/BasicLTI.xml I_3995af9f-819b-3359-a0df-58380a566eb5_3_R/BasicLTI.xml I_39966eec-dca2-3970-84da-a32871b53497_R/BasicLTI.xml I_399e3be5-d40e-3c7b-a605-949ae4d31c11_1_R/BasicLTI.xml I_399e3be5-d40e-3c7b-a605-949ae4d31c11_3_R/BasicLTI.xml I_399e3be5-d40e-3c7b-a605-949ae4d31c11_5_R/BasicLTI.xml I_39b220d8-9e4b-3b16-9a83-ae16aa1f25fb_R/BasicLTI.xml I_39bf797d-7fa5-3a0a-b5f8-b53871fcc2c0_1_R/BasicLTI.xml I_3a26fcd1-bdb8-303a-a6ae-14f1d5a243c8_1_R/BasicLTI.xml I_3a52392c-4fda-311a-9b98-e06df0c2c35e_1_R/BasicLTI.xml I_3a8788de-1b8a-35c6-9755-29f2b6f19ebb_1_R/BasicLTI.xml I_3aa62e15-4340-3386-bdd7-6dc0e513040c_R/BasicLTI.xml I_3aaa9bd7-5686-3b1c-b43b-9a50c598bbf7_R/BasicLTI.xml I_3ab9843d-abd3-3e4f-9219-e7cac7ba85d6_R/BasicLTI.xml I_3ad1fe30-fa01-3084-b0b6-c08dda919db3_R/BasicLTI.xml I_3adc9881-eb22-35e5-8065-a867937d9603_1_R/BasicLTI.xml I_3adc9881-eb22-35e5-8065-a867937d9603_3_R/BasicLTI.xml I_3adc9881-eb22-35e5-8065-a867937d9603_5_R/BasicLTI.xml I_3adc9881-eb22-35e5-8065-a867937d9603_7_R/BasicLTI.xml I_3addc8dc-ce0d-3864-a9ed-4423aa512fc9_R/BasicLTI.xml I_3afdd34b-7e1d-3564-8c7c-f81c39e60f1a_1_R/BasicLTI.xml I_3aff8353-7bd7-3e48-a063-7a10b7b29f98_R/BasicLTI.xml I_3b21b862-f89b-35f8-9cb6-02de8567c062_1_R/BasicLTI.xml I_3b511d65-82cf-3319-a253-c70200b9d384_R/BasicLTI.xml I_3b62e137-9235-3033-8bc3-6f82f1b86a47_R/BasicLTI.xml I_3b6bcc94-a2d4-3e9d-b93d-e98592501a84_1_R/BasicLTI.xml I_3b6d3b9d-f43c-390d-b206-8f4d86353295_1_R/BasicLTI.xml I_3b965154-2073-3f1b-bacc-cd5a419eefed_1_R/BasicLTI.xml I_3b965c4c-63e3-3816-8e94-3cc9e0b3c8c3_1_R/BasicLTI.xml I_3ba01a60-bbf5-346f-a53d-e582f02e5077_1_R/BasicLTI.xml I_3ba01a60-bbf5-346f-a53d-e582f02e5077_3_R/BasicLTI.xml I_3ba01a60-bbf5-346f-a53d-e582f02e5077_5_R/BasicLTI.xml I_3ba01a60-bbf5-346f-a53d-e582f02e5077_7_R/BasicLTI.xml I_3ba23d57-f113-3295-bced-13d78897ed95_R/BasicLTI.xml I_3bb2955d-ca66-30c1-b04a-202fa2645361_R/BasicLTI.xml I_3bb453d2-e336-332e-af0e-8690b0771a9c_1_R/BasicLTI.xml I_3bb453d2-e336-332e-af0e-8690b0771a9c_3_R/BasicLTI.xml I_3bb453d2-e336-332e-af0e-8690b0771a9c_5_R/BasicLTI.xml I_3bb453d2-e336-332e-af0e-8690b0771a9c_7_R/BasicLTI.xml I_3bb89291-764b-3aca-8390-718f11c15fde_1_R/BasicLTI.xml I_3bd200fc-0d62-334c-8d8d-95e232da9b9b_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_11_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_13_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_1_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_3_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_5_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_7_R/BasicLTI.xml I_3bd89d44-5288-3911-876e-e1376460cbf8_9_R/BasicLTI.xml I_3bfff8b7-df0f-3a32-a880-eba380757292_1_R/BasicLTI.xml I_3c159983-0321-383e-9e5b-ee8cc45b62dd_R/BasicLTI.xml I_3c3ca39b-9711-3c78-8f2a-b4c94a625cf9_1_R/BasicLTI.xml I_3c55894d-c660-3890-bb59-f95f6d8b4293_1_R/BasicLTI.xml I_3c598fe8-46eb-318d-80f0-2979034855d9_1_R/BasicLTI.xml I_3c5bf539-6fa7-3df5-a9b5-dfc1739dd1cd_1_R/BasicLTI.xml I_3c7ae04d-a1ac-360e-8945-480f7c18a932_R/BasicLTI.xml I_3c82c08f-7d30-3f2d-8695-c4d3878496d8_1_R/BasicLTI.xml I_3c96d979-4d22-33a4-9377-b9cf6a718fce_1_R/BasicLTI.xml I_3ca72d6b-edcf-35c3-ad0f-e14aedb578a2_R/BasicLTI.xml I_3cb1b5d5-20b6-36bb-846f-c30e96cd9a4e_1_R/BasicLTI.xml I_3cb1b5d5-20b6-36bb-846f-c30e96cd9a4e_3_R/BasicLTI.xml I_3cb1b5d5-20b6-36bb-846f-c30e96cd9a4e_5_R/BasicLTI.xml I_3cb1b5d5-20b6-36bb-846f-c30e96cd9a4e_7_R/BasicLTI.xml I_3cd8bba3-2daf-3544-b085-a9efb7f3bd0f_1_R/BasicLTI.xml I_3cd8bba3-2daf-3544-b085-a9efb7f3bd0f_3_R/BasicLTI.xml I_3cebb160-fb56-3c6e-babb-13d9ba2d1d43_1_R/BasicLTI.xml I_3cfee524-dde6-3730-b5fd-d8758e31293e_R/BasicLTI.xml I_3d0fc782-34dd-3c0a-98a5-ffd58ee09296_1_R/BasicLTI.xml I_3d25bbd5-e4ba-3e84-8a90-503ca49b52e8_R/BasicLTI.xml I_3d3b158c-6e2e-3a65-b7d4-ef8225e4e284_1_R/BasicLTI.xml I_3d3e7bc8-cc38-37ba-8249-e35fac4efecb_1_R/BasicLTI.xml I_3d3e7bc8-cc38-37ba-8249-e35fac4efecb_3_R/BasicLTI.xml I_3d7b5026-5bba-3202-9dde-c74420190b3d_1_R/BasicLTI.xml I_3da41ed2-f5b4-3ded-a849-b5981a5aff4f_1_R/BasicLTI.xml I_3da41ed2-f5b4-3ded-a849-b5981a5aff4f_3_R/BasicLTI.xml I_3da41ed2-f5b4-3ded-a849-b5981a5aff4f_5_R/BasicLTI.xml I_3db7ac0a-5422-3bd9-a289-d3178c07bab8_1_R/BasicLTI.xml I_3db7ff76-590d-3540-a86d-84c45255b332_R/BasicLTI.xml I_3dc65494-2e36-30ad-ac22-e73aff80cbfa_R/BasicLTI.xml I_3de7207e-6dfb-36d5-a894-d501fa7313a6_1_R/BasicLTI.xml I_3de7207e-6dfb-36d5-a894-d501fa7313a6_3_R/BasicLTI.xml I_3de7207e-6dfb-36d5-a894-d501fa7313a6_5_R/BasicLTI.xml I_3de7207e-6dfb-36d5-a894-d501fa7313a6_7_R/BasicLTI.xml I_3dfbca14-6b75-3f8f-b211-71507a05f49c_1_R/BasicLTI.xml I_3dfec499-a4eb-37ad-b832-79e2f535e527_R/BasicLTI.xml I_3e0211cf-f4c0-38e0-92ac-1cd2b3343a78_R/BasicLTI.xml I_3e06cc13-f66d-3f82-bc72-0e1fb5692dbe_R/BasicLTI.xml I_3e13cea1-97c1-3e2a-b7e0-667e3891ef7b_1_R/BasicLTI.xml I_3e209d24-8832-3157-aac7-3c672ab314dd_1_R/BasicLTI.xml I_3e2f8e6f-df17-35c5-8634-ecb4c03aae9a_1_R/BasicLTI.xml I_3e3bb0d0-fdfe-355a-a1dd-8c528ab526d0_R/BasicLTI.xml I_3e3cb882-c37a-3cf9-ad60-060a555420b7_1_R/BasicLTI.xml I_3e3cb882-c37a-3cf9-ad60-060a555420b7_3_R/BasicLTI.xml I_3e3cb882-c37a-3cf9-ad60-060a555420b7_5_R/BasicLTI.xml I_3e528c27-6522-34cf-b1d4-872ee3d67fe8_1_R/BasicLTI.xml I_3e528c27-6522-34cf-b1d4-872ee3d67fe8_3_R/BasicLTI.xml I_3e528c27-6522-34cf-b1d4-872ee3d67fe8_5_R/BasicLTI.xml I_3e6600c0-c42d-3cac-a049-88b14b9840a0_1_R/BasicLTI.xml I_3e797e47-f5de-357c-9b58-f8bbab6f9384_1_R/BasicLTI.xml I_3e884523-20be-3c66-a700-60abc9cd1339_1_R/BasicLTI.xml I_3e8c21b1-5a1c-3feb-bb0c-74b7c2dd954a_R/BasicLTI.xml I_3e929e4a-017a-3683-a0cf-94c2d96ee86b_1_R/BasicLTI.xml I_3ea59eaf-43c4-33df-93fa-d11b965cc1e8_R/BasicLTI.xml I_3ea9ed13-0a0a-3b6a-9969-1babc73f24c0_1_R/BasicLTI.xml I_3ea9f948-9e94-38fe-8e46-162598240b5a_1_R/BasicLTI.xml I_3ed93257-a136-3014-86ce-c87f4c7f290c_1_R/BasicLTI.xml I_3ee1cc97-3307-303e-be6d-97976a791366_1_R/BasicLTI.xml I_3ef4894d-778e-3c60-9242-65f13e9de43f_1_R/BasicLTI.xml I_3efe40e6-8ae9-3aac-98c9-9c0ae2bf785d_R/BasicLTI.xml I_3f048aa4-6d0d-3b9c-8b49-956d3e2a3bfe_1_R/BasicLTI.xml I_3f048aa4-6d0d-3b9c-8b49-956d3e2a3bfe_3_R/BasicLTI.xml I_3f373670-dc1e-31d6-ba99-281e5e9d2d10_1_R/BasicLTI.xml I_3f464375-2ac4-370e-a8d0-d29d6c594e82_1_R/BasicLTI.xml I_3f464375-2ac4-370e-a8d0-d29d6c594e82_3_R/BasicLTI.xml I_3f464375-2ac4-370e-a8d0-d29d6c594e82_5_R/BasicLTI.xml I_3f464375-2ac4-370e-a8d0-d29d6c594e82_7_R/BasicLTI.xml I_3f464375-2ac4-370e-a8d0-d29d6c594e82_9_R/BasicLTI.xml I_3fa8dc41-9019-321a-aeb9-232d5aa11c42_R/BasicLTI.xml I_3fc0ad3c-71ce-3d20-879f-ffea2cae5cb5_1_R/BasicLTI.xml I_3fce7173-04e4-33e5-96e3-8d414f7a2ddc_R/BasicLTI.xml I_4018ed4b-c10c-3ba6-960a-ea49d65178c8_1_R/BasicLTI.xml I_403a1621-cf08-31b8-b5a0-f13cdbf8a92f_R/BasicLTI.xml I_404c0a84-23d3-3523-8654-53daf91a9d68_1_R/BasicLTI.xml I_40739d98-ee9d-3532-881b-dc3938c2442e_1_R/BasicLTI.xml I_40d4ce8d-cc3e-3a4a-b93f-5ab6437d3c95_1_R/BasicLTI.xml I_410996e5-3ffa-39fb-84a9-f3cb2a69cc88_R/BasicLTI.xml I_41205fa3-74c4-3f7d-988b-84718f0c9a4d_R/BasicLTI.xml I_41315a99-8675-3f37-a060-3c8d750ae064_1_R/BasicLTI.xml I_414177e2-90b6-385e-9efe-c25f9a2d17f7_1_R/BasicLTI.xml I_41431adc-2d51-38bd-a093-6395be54f836_1_R/BasicLTI.xml I_4150890f-84aa-3ef8-ba4d-33b672fbd2d8_1_R/BasicLTI.xml I_4150b184-4a42-3885-8a6c-72fba2286816_R/BasicLTI.xml I_415bf556-9070-3fab-84af-685e3a4889a1_1_R/BasicLTI.xml I_41b07434-02f2-3f16-a0b0-7c997b99df52_R/BasicLTI.xml I_41e87779-39d9-39b2-939e-ce957c7cbcdc_1_R/BasicLTI.xml I_41ec46a8-c37e-3789-bd4d-1e358c5c4f3f_1_R/BasicLTI.xml I_4256cce3-9c1a-3e8e-a756-44d90448ed5e_R/BasicLTI.xml I_425d9ee9-463a-3f79-8ed6-b808db91bbd3_1_R/BasicLTI.xml I_425f589e-d76d-3e91-96c3-59306828504c_1_R/BasicLTI.xml I_427a7b1f-908a-35f8-842c-8c5dbca3bca9_1_R/BasicLTI.xml I_427c6270-e7a9-3ae7-b6fb-30ee9cf424e2_1_R/BasicLTI.xml I_42801399-7b0d-3ca2-b4ef-b95cf2f3b9d4_R/BasicLTI.xml I_428a0877-65d5-3137-986d-7a54daa3f1e8_1_R/BasicLTI.xml I_428a0877-65d5-3137-986d-7a54daa3f1e8_3_R/BasicLTI.xml I_428a0877-65d5-3137-986d-7a54daa3f1e8_5_R/BasicLTI.xml I_428a0877-65d5-3137-986d-7a54daa3f1e8_7_R/BasicLTI.xml I_429180c3-337c-38aa-8abc-74cc7af82c00_R/BasicLTI.xml I_4299d56a-1c61-326a-be97-2c2b8bd77358_1_R/BasicLTI.xml I_429fb3b8-ac65-3997-a6a7-fa71d07df7f3_1_R/BasicLTI.xml I_42b49016-c954-3912-9d01-0cb7b904525a_1_R/BasicLTI.xml I_42d747e6-a5ac-365b-96c8-f0cea3065072_1_R/BasicLTI.xml I_42d7a2de-26f5-3f4f-ac94-b05d221dd9c2_1_R/BasicLTI.xml I_43065314-5de3-3259-af52-9378e813ecad_R/BasicLTI.xml I_4319452f-d19e-3116-bef5-95232e14f90a_1_R/BasicLTI.xml I_43423444-cb52-3626-b526-234385d4e717_1_R/BasicLTI.xml I_4342f023-6cf1-3350-9c99-e063d4883409_1_R/BasicLTI.xml I_4342f023-6cf1-3350-9c99-e063d4883409_3_R/BasicLTI.xml I_4342f023-6cf1-3350-9c99-e063d4883409_5_R/BasicLTI.xml I_4342f023-6cf1-3350-9c99-e063d4883409_7_R/BasicLTI.xml I_4342f023-6cf1-3350-9c99-e063d4883409_9_R/BasicLTI.xml I_43583bc7-e882-353c-b815-9f4667284ecd_1_R/BasicLTI.xml I_437c2ab9-1c93-3206-8b22-974c26fa44d3_1_R/BasicLTI.xml I_4389656b-2857-3809-9874-f7522e5e52bf_R/BasicLTI.xml I_43a2f4b1-3712-31b4-8176-18cceba2750f_R/BasicLTI.xml I_43a3618a-8d45-3e43-8140-df8d14a52757_1_R/BasicLTI.xml I_43a8e98e-aa42-3c55-88a8-5076512d100e_1_R/BasicLTI.xml I_43b7448a-2265-3da3-941d-8f03492625f9_1_R/BasicLTI.xml I_43e0be0f-cb6c-3106-87be-95890ddde1c8_R/BasicLTI.xml I_43ee991c-22a5-31a7-bb5d-214f86825897_1_R/BasicLTI.xml I_43f59c14-a368-3bc0-85ed-781c8b361e5a_1_R/BasicLTI.xml I_43fee280-8d60-326c-9adc-0b14f91c7c41_1_R/BasicLTI.xml I_43fee280-8d60-326c-9adc-0b14f91c7c41_3_R/BasicLTI.xml I_440dacec-d54a-3fd2-9b42-e3580f688edf_1_R/BasicLTI.xml I_44259308-62f8-3f69-be7c-3b62d763cb52_1_R/BasicLTI.xml I_44259308-62f8-3f69-be7c-3b62d763cb52_3_R/BasicLTI.xml I_44404391-061d-3c2e-bbb0-9e770f34e1e0_1_R/BasicLTI.xml I_444c22b8-d9c4-3c59-b61d-7dbee773266e_1_R/BasicLTI.xml I_4493d27d-8fe0-315f-ad91-c8408a6a821e_R/BasicLTI.xml I_44a3f05d-fb08-31b6-904a-be273e1d65fb_R/BasicLTI.xml I_44d169b1-3e30-3c0e-8bab-3760e880c21e_1_R/BasicLTI.xml I_44fbc4e4-a8f6-3825-9894-3de0ef907731_R/BasicLTI.xml I_4506fb24-7ec3-38b6-a972-e57db9945ca3_1_R/BasicLTI.xml I_4516e749-f116-34f2-a8ac-7a54bf20dc40_R/BasicLTI.xml I_452b82c1-e1ad-31bb-b353-5ed5c71df643_1_R/BasicLTI.xml I_452b82c1-e1ad-31bb-b353-5ed5c71df643_3_R/BasicLTI.xml I_4555ef2a-be15-3d8d-8676-ad9cc924485d_1_R/BasicLTI.xml I_4557bbd6-9993-3d91-af11-8aa607dd966c_R/BasicLTI.xml I_45745f17-cca4-3617-ab80-4f0299d277b5_R/BasicLTI.xml I_45a4b513-5587-378c-87b8-2e4f1c5bf988_R/BasicLTI.xml I_45a98797-c65b-3bd1-801d-9e2b12b05102_1_R/BasicLTI.xml I_45a98797-c65b-3bd1-801d-9e2b12b05102_3_R/BasicLTI.xml I_45a98797-c65b-3bd1-801d-9e2b12b05102_5_R/BasicLTI.xml I_45a98797-c65b-3bd1-801d-9e2b12b05102_7_R/BasicLTI.xml I_45ae9c54-7cfa-3a79-99d4-c852ce50e62e_R/BasicLTI.xml I_45bc7acc-1ac1-3263-aaff-ef8d50fdeee3_R/BasicLTI.xml I_45beeab5-708a-32ef-8579-17867a2147d4_R/BasicLTI.xml I_45c7e416-ca08-3252-9f7a-accd2c115286_1_R/BasicLTI.xml I_45c826ca-e330-3043-8a17-e4e18fd2798a_1_R/BasicLTI.xml I_45c826ca-e330-3043-8a17-e4e18fd2798a_3_R/BasicLTI.xml I_45c826ca-e330-3043-8a17-e4e18fd2798a_5_R/BasicLTI.xml I_45c826ca-e330-3043-8a17-e4e18fd2798a_7_R/BasicLTI.xml I_45d5054e-e12e-317a-8f73-be591cde7c53_1_R/BasicLTI.xml I_45d85d2c-4a07-348b-a213-7189d945694a_R/BasicLTI.xml I_45e513b2-8d3c-37ae-aafc-849886fb20cd_1_R/BasicLTI.xml I_45e513b2-8d3c-37ae-aafc-849886fb20cd_3_R/BasicLTI.xml I_45e79185-bda2-314c-82bc-9b7203890d6f_1_R/BasicLTI.xml I_45f68b0f-d387-3d2f-a393-ac2ebd79b442_R/BasicLTI.xml I_4609b03b-b2dc-3811-a187-a88f7bc86d48_1_R/BasicLTI.xml I_4609b03b-b2dc-3811-a187-a88f7bc86d48_3_R/BasicLTI.xml I_46126417-1b37-3eef-bf98-53d0ceebebc2_R/BasicLTI.xml I_46296725-ed81-3f36-bfb6-3c1e006722de_R/BasicLTI.xml I_4651ed16-8185-3afe-b611-b0fd6f209a9c_1_R/BasicLTI.xml I_465ad7b5-031f-3f71-aa50-a426d130c0ee_1_R/BasicLTI.xml I_465ad7b5-031f-3f71-aa50-a426d130c0ee_3_R/BasicLTI.xml I_465ad7b5-031f-3f71-aa50-a426d130c0ee_5_R/BasicLTI.xml I_465ad7b5-031f-3f71-aa50-a426d130c0ee_7_R/BasicLTI.xml I_465c7161-5c38-3f58-bf2b-fda309a12f96_1_R/BasicLTI.xml I_466d8699-b90c-3be8-adda-e1df0c208256_1_R/BasicLTI.xml I_46916c97-6f0d-3bcb-aa4c-86cbcdf8fe6e_1_R/BasicLTI.xml I_46969014-2e38-32c6-8046-abfb88d9ca71_R/BasicLTI.xml I_46a54e0c-9fab-3d17-82bd-751e4e440ed8_R/BasicLTI.xml I_46b841d9-96aa-3c7d-9e1f-60b247b27000_R/BasicLTI.xml I_46bb9a9a-4664-39b9-a32d-729f09b55f06_1_R/BasicLTI.xml I_46bb9a9a-4664-39b9-a32d-729f09b55f06_3_R/BasicLTI.xml I_46bb9a9a-4664-39b9-a32d-729f09b55f06_5_R/BasicLTI.xml I_46bb9a9a-4664-39b9-a32d-729f09b55f06_7_R/BasicLTI.xml I_46e81261-1e9f-3495-b36e-bda2a7bba0b1_1_R/BasicLTI.xml I_46e81261-1e9f-3495-b36e-bda2a7bba0b1_3_R/BasicLTI.xml I_46ec2991-9861-3a57-b3de-05df064af2de_R/BasicLTI.xml I_470c9eff-8384-3c38-8452-508b37267a0e_R/BasicLTI.xml I_4713f197-eaf8-3275-ab25-322b9cbb5730_1_R/BasicLTI.xml I_471789d6-3a28-336e-ae36-c729600761a3_1_R/BasicLTI.xml I_471ac75a-a457-3332-8ba6-a904054608fa_1_R/BasicLTI.xml I_471ac75a-a457-3332-8ba6-a904054608fa_3_R/BasicLTI.xml I_471e5399-23b4-3177-979b-7f31a85e8e7e_1_R/BasicLTI.xml I_471e5399-23b4-3177-979b-7f31a85e8e7e_3_R/BasicLTI.xml I_471e5399-23b4-3177-979b-7f31a85e8e7e_5_R/BasicLTI.xml I_471e5399-23b4-3177-979b-7f31a85e8e7e_7_R/BasicLTI.xml I_471f1d9f-0441-3593-836e-509787633db1_R/BasicLTI.xml I_4722ab0a-0e3a-309b-a1c6-83ccc7186b45_1_R/BasicLTI.xml I_47308f5d-0eaa-3833-970c-8b516a8ebf4b_R/BasicLTI.xml I_47695160-d567-3a8e-9cb6-8f225ec5dba0_1_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_11_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_1_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_3_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_5_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_7_R/BasicLTI.xml I_476ea55e-8233-3d37-880b-efac9a4ef973_9_R/BasicLTI.xml I_477966ff-6614-3870-b811-beb1b5d02fa2_R/BasicLTI.xml I_47ad8f8c-4797-3dd4-9e42-de7877c0f129_R/BasicLTI.xml I_47b8aa7c-53f2-3c66-9e2d-443f841d3145_R/BasicLTI.xml I_47faad64-6302-3895-90cc-3d78d9a12ef7_1_R/BasicLTI.xml I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_1_R/BasicLTI.xml I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_3_R/BasicLTI.xml I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_5_R/BasicLTI.xml I_481416b4-5cb6-3e9c-a117-8c5910e26cf8_7_R/BasicLTI.xml I_481ae19c-bbc2-3f28-b47e-96a8bccc6faf_R/BasicLTI.xml I_4831b740-783d-3e84-a002-b9defc4aa15d_1_R/BasicLTI.xml I_4831b740-783d-3e84-a002-b9defc4aa15d_3_R/BasicLTI.xml I_4831b740-783d-3e84-a002-b9defc4aa15d_5_R/BasicLTI.xml I_484696e2-a79d-3a4e-91a2-d1250075cf41_1_R/BasicLTI.xml I_48619964-91ea-3121-9a61-9a7861cda3ec_1_R/BasicLTI.xml I_487dd3c3-c85e-347b-b724-04943ab092f1_1_R/BasicLTI.xml I_4894c9e0-5e7a-36d7-bed2-d018c08304d1_1_R/BasicLTI.xml I_48af29fe-41a2-35d9-b375-09e7a3968ee9_1_R/BasicLTI.xml I_48d6ad6e-1dd3-3486-b25e-df12ca5893a4_R/BasicLTI.xml I_4903bac0-6895-3b73-a3f9-aaf425d0ad57_1_R/BasicLTI.xml I_491331e7-36c8-3990-9e3a-2b38a70185cc_R/BasicLTI.xml I_491689ef-944d-32c2-aa5c-12e5bda862eb_1_R/BasicLTI.xml I_49181a75-bbdd-3ae6-98f6-02869036f39d_R/BasicLTI.xml I_4920733f-70dc-3ea0-9ed5-d2b6045b881c_R/BasicLTI.xml I_49279c8c-fed3-3f76-a543-eaf4bc5763c2_1_R/BasicLTI.xml I_4935c4b0-0585-33d7-81d1-5623d384ec0c_R/BasicLTI.xml I_4939889f-272e-3e69-8afb-591d6da0bcd3_1_R/BasicLTI.xml I_493cf852-3a72-3741-b59e-86b4ddde62f4_1_R/BasicLTI.xml I_4980332b-9154-3c0f-b0e0-774d6995847b_1_R/BasicLTI.xml I_498dd2e9-7b5b-3fec-bce6-e8fa2fe85e9b_1_R/BasicLTI.xml I_499d0bdf-70a5-3d9e-83f1-efe2646cdb3a_R/BasicLTI.xml I_49a69134-d399-3cdd-8b3c-d0cab6ae13ac_R/BasicLTI.xml I_49d17458-ad6c-34ef-b4af-7d8483e38d14_1_R/BasicLTI.xml I_49db8e10-5708-3264-b876-e786f80a70aa_R/BasicLTI.xml I_49e5d12f-875d-3dc8-8424-cef8b27de0c8_1_R/BasicLTI.xml I_49ed8429-bdd1-3f60-820b-5a2614e9e235_1_R/BasicLTI.xml I_49ed8429-bdd1-3f60-820b-5a2614e9e235_3_R/BasicLTI.xml I_49f67dee-aced-3c99-90da-c41af6577c1f_1_R/BasicLTI.xml I_4a05ab33-f529-3bec-800e-4182830b7ac1_1_R/BasicLTI.xml I_4a2d1e63-1feb-3ebd-9c76-373d62376ad0_1_R/BasicLTI.xml I_4a322989-d973-3620-b4ad-eee7a8323823_R/BasicLTI.xml I_4a430634-c531-37a3-893e-1282a8fed7ad_1_R/BasicLTI.xml I_4a430634-c531-37a3-893e-1282a8fed7ad_3_R/BasicLTI.xml I_4a6fead1-2ca0-31a6-ada3-aaa09eb22ce7_1_R/BasicLTI.xml I_4a70eb77-681a-3454-9a3c-240d584b05d4_1_R/BasicLTI.xml I_4a75d936-c926-3449-82ad-efea3b8be4f8_R/BasicLTI.xml I_4a8a5225-6f30-3a63-ba4a-08cfd15f3352_1_R/BasicLTI.xml I_4a98bb81-d193-3a8a-9509-4864d7cc472b_R/BasicLTI.xml I_4a9eac37-2fc1-371c-a449-fc068a0ee264_1_R/BasicLTI.xml I_4aa1a01d-50e5-39b9-bc42-751d3171f2b3_1_R/BasicLTI.xml I_4ac2d984-fe9d-3bee-912e-b26c1a0dcd46_1_R/BasicLTI.xml I_4ac3a347-0b41-3c0f-a3fb-ece194b2f899_1_R/BasicLTI.xml I_4ad98da3-b57e-3654-aa0c-b1909df83847_1_R/BasicLTI.xml I_4ad98da3-b57e-3654-aa0c-b1909df83847_3_R/BasicLTI.xml I_4aecd205-ab6c-370d-baef-d894b85992d5_R/BasicLTI.xml I_4afe60b0-1b41-3a2f-9e0a-76d07de0e0c1_R/BasicLTI.xml I_4b055a6a-0614-3e18-8bc1-e91fb0ccae02_1_R/BasicLTI.xml I_4b1f45ea-cafe-398e-94d6-dcd4f7d048f5_1_R/BasicLTI.xml I_4b4bbb4d-c34e-3f31-bce1-fe0a9e6e9506_1_R/BasicLTI.xml I_4b74196f-0c38-35bb-8a9d-f7ecd522b0fc_R/BasicLTI.xml I_4b7fa1eb-fd09-3d21-8714-4f775da49f10_1_R/BasicLTI.xml I_4b7fa1eb-fd09-3d21-8714-4f775da49f10_3_R/BasicLTI.xml I_4b7fa1eb-fd09-3d21-8714-4f775da49f10_5_R/BasicLTI.xml I_4b843a82-3cd0-3aab-bd47-50ed1609a4eb_1_R/BasicLTI.xml I_4b843a82-3cd0-3aab-bd47-50ed1609a4eb_3_R/BasicLTI.xml I_4b843a82-3cd0-3aab-bd47-50ed1609a4eb_5_R/BasicLTI.xml I_4b843a82-3cd0-3aab-bd47-50ed1609a4eb_7_R/BasicLTI.xml I_4b992a70-0ba6-3863-91b0-1c0ebc07ba36_1_R/BasicLTI.xml I_4b9d77ea-36ba-3de6-a957-e097674e7c16_1_R/BasicLTI.xml I_4bbc6b85-7dbb-3d63-83ae-96fd9ab5e2a2_1_R/BasicLTI.xml I_4bf09c8c-cce3-3585-901c-f2afaaf70fa5_R/BasicLTI.xml I_4bf41f30-075d-3191-91e6-fd67ca799db4_1_R/BasicLTI.xml I_4bfba323-0f12-3700-93b0-e1a37a990ce9_1_R/BasicLTI.xml I_4c070bcf-57f8-375a-8c62-d629c2c6a4af_R/BasicLTI.xml I_4c0c2baf-eb57-32eb-b03e-212d640a8e9f_R/BasicLTI.xml I_4c1bbaf0-48a6-3b83-a31b-3f8014a5ebec_R/BasicLTI.xml I_4c2a7cdf-ca46-373d-8df4-ec68831fb9c2_R/BasicLTI.xml I_4c6d8d30-b917-3c20-aeb5-c76bfc71e98d_1_R/BasicLTI.xml I_4c79195b-2c4a-3c78-be05-2ce07e711200_R/BasicLTI.xml I_4c81bc4b-907f-35cb-a3fc-84a389b73d4d_1_R/BasicLTI.xml I_4c81bc4b-907f-35cb-a3fc-84a389b73d4d_3_R/BasicLTI.xml I_4c820ed7-8139-326a-ad0a-fccc64fc387c_1_R/BasicLTI.xml I_4c8982b9-3441-3de5-b9cf-04ae63592e6f_R/BasicLTI.xml I_4caba029-ec36-30a1-a455-fc101cda0dc9_1_R/BasicLTI.xml I_4cb3b240-8e24-32fa-9c88-e871cc13c37c_1_R/BasicLTI.xml I_4cba2ba1-73bf-33f6-9315-0d9e7c0cc906_1_R/BasicLTI.xml I_4cdbd4ac-4693-3c78-86a6-44dd443bdc66_R/BasicLTI.xml I_4ce7fd28-e5d7-36a5-ab1a-4890a9cd130c_1_R/BasicLTI.xml I_4cf35d9b-fbc1-3f68-8b79-cc6733f7d213_1_R/BasicLTI.xml I_4d03f8a2-5b8d-3ce3-a03c-92f720a630d1_1_R/BasicLTI.xml I_4d1e1f11-59d7-3f31-909d-54cdc7e47d41_R/BasicLTI.xml I_4d20dea6-9b96-39ad-91db-a4e84f5e0d93_1_R/BasicLTI.xml I_4d270e20-5cbe-3269-871b-11bfb85c66db_R/BasicLTI.xml I_4d2cd2a8-b55f-300a-8311-61097aba2d94_R/BasicLTI.xml I_4d3f0da4-d59a-3019-b3f8-bf8ea739ed6f_R/BasicLTI.xml I_4d40d974-a0a1-3290-8259-acc29e3c0a29_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_11_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_1_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_3_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_5_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_7_R/BasicLTI.xml I_4d5509e5-da16-3f27-a7bc-f9ce2759cd7c_9_R/BasicLTI.xml I_4d5c938d-029c-3520-b98f-7d86803e389a_R/BasicLTI.xml I_4d754b02-9e18-374b-a17d-149cea3466dc_1_R/BasicLTI.xml I_4d8d95b8-5595-3cb7-bac9-4bc5a2116854_1_R/BasicLTI.xml I_4d93047a-8a00-3622-9c6b-f1d21e8d529b_1_R/BasicLTI.xml I_4d9a7d92-ca9e-34c0-8b79-993d022112d4_R/BasicLTI.xml I_4da63258-18aa-32fb-a4a1-2e6333e15c06_R/BasicLTI.xml I_4dbee59a-154c-3416-9cee-c327aaaaa3cb_R/BasicLTI.xml I_4dd11135-342e-3e35-b210-49074f54aa28_1_R/BasicLTI.xml I_4ddee176-8277-31b6-a2bc-1c7d9bc22d58_1_R/BasicLTI.xml I_4ddf7aef-ef9d-3e68-8132-12c3068d7876_1_R/BasicLTI.xml I_4ddfd98b-a236-32c9-bf45-131dee22096e_1_R/BasicLTI.xml I_4e15112b-786a-310b-9fd4-55a4ef4286fb_R/BasicLTI.xml I_4e162aa7-043a-3f39-8b01-5b48d42d6db4_R/BasicLTI.xml I_4e22d3bd-3ba7-397d-a543-822c5200ad54_R/BasicLTI.xml I_4e26cc21-405b-32da-af96-b2c2e8022c27_1_R/BasicLTI.xml I_4e3affc9-feb8-3326-8e64-0e7f9babbb98_R/BasicLTI.xml I_4e4d553b-4358-398c-8f24-0097aaf996eb_R/BasicLTI.xml I_4e58c9cc-57e1-3bb8-930d-c2e22ef9ae47_1_R/BasicLTI.xml I_4e70dc75-dee2-365e-98f5-105ac4a10c30_1_R/BasicLTI.xml I_4e798345-7991-3a94-b59b-c42e138c601a_1_R/BasicLTI.xml I_4e798345-7991-3a94-b59b-c42e138c601a_3_R/BasicLTI.xml I_4e798345-7991-3a94-b59b-c42e138c601a_5_R/BasicLTI.xml I_4e7c95f2-a85d-39b9-a68a-4ed664393e0c_1_R/BasicLTI.xml I_4e824c7d-5574-3316-9239-2b0d975ac4d9_1_R/BasicLTI.xml I_4e914694-96b6-3025-a2bc-e1c07e8634ce_R/BasicLTI.xml I_4e9bd9f5-9c60-3afc-931f-290d524b99bd_1_R/BasicLTI.xml I_4ea68edb-5d61-3329-9edf-0ab426cf63ee_1_R/BasicLTI.xml I_4eb06690-d2b6-3b29-99bf-e11fb6d4cf50_1_R/BasicLTI.xml I_4eb1b9e7-a1a8-3b22-8714-df204b37f6d5_1_R/BasicLTI.xml I_4ed9cb4f-3358-37fd-96f2-722c4b623f60_R/BasicLTI.xml I_4ee804da-7195-3bf6-ab42-e8cb5970ff49_R/BasicLTI.xml I_4ef45d86-48af-3d7e-96b5-8c2deaec1e72_R/BasicLTI.xml I_4ef7638f-13b6-380a-9cc1-52460b5b2eb9_R/BasicLTI.xml I_4f0939d9-2bdb-356c-8389-0bd00414ac6a_1_R/BasicLTI.xml I_4f353a4b-8631-3994-bc38-f7a4c2852620_1_R/BasicLTI.xml I_4f39ffb9-0617-3713-8e56-e50cc636eb9f_R/BasicLTI.xml I_4f4a6646-8561-311b-8eec-bf573801bfd1_R/BasicLTI.xml I_4f5d9c2f-ee32-304b-b547-9724cb0dde19_1_R/BasicLTI.xml I_4f673d46-13c7-324b-b860-a4ba576f4f37_1_R/BasicLTI.xml I_4f6d6f0a-5f12-3b0a-bc35-31eb92eca263_1_R/BasicLTI.xml I_4f799fe9-9d54-3511-8834-88d4598af7d2_1_R/BasicLTI.xml I_4f9be200-1608-3236-821f-7d03e496cd2e_1_R/BasicLTI.xml I_4fb88c72-c1c4-3903-9aaa-7ccd73a28e89_1_R/BasicLTI.xml I_50046e89-cce3-3198-96fe-05073daa01d0_1_R/BasicLTI.xml I_50348808-6f4a-3824-bdd5-332cc50e2385_1_R/BasicLTI.xml I_503fe5b9-3ed7-3dcb-85be-ae5302316422_1_R/BasicLTI.xml I_5040b26a-bf36-3d82-8e2a-dd6bb36ec5d1_1_R/BasicLTI.xml I_5078ac3a-5506-3889-8df2-386a1a9eeeea_1_R/BasicLTI.xml I_507909a4-fb3d-3a3b-9d10-05a0213a9ff2_R/BasicLTI.xml I_507bab23-7cbc-30dc-acf6-ce102a04a1a5_R/BasicLTI.xml I_507dbe84-d840-34ce-9706-9726988190d4_1_R/BasicLTI.xml I_507f4bd5-a009-305b-af2f-3dfc0e5ae02b_R/BasicLTI.xml I_5089901c-5573-39aa-96b2-4c3926d1fcab_1_R/BasicLTI.xml I_5096ab28-31ec-3558-9c56-8590bea8a172_R/BasicLTI.xml I_509b5f13-a6bd-37d2-b98f-5c34f0640395_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_11_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_13_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_15_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_1_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_3_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_5_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_7_R/BasicLTI.xml I_50d8cb51-8b71-3451-a732-90b29022a903_9_R/BasicLTI.xml I_50e879f7-2eb7-3a05-b0ab-8b4a5d87e132_R/BasicLTI.xml I_51124a14-100d-36cd-a538-e427d5e92378_1_R/BasicLTI.xml I_5120c48d-6b45-3328-b3c6-79781f581f2a_1_R/BasicLTI.xml I_5120c48d-6b45-3328-b3c6-79781f581f2a_3_R/BasicLTI.xml I_5120c48d-6b45-3328-b3c6-79781f581f2a_5_R/BasicLTI.xml I_5120c48d-6b45-3328-b3c6-79781f581f2a_7_R/BasicLTI.xml I_51303acf-feda-3d1a-985e-2224215cbe37_1_R/BasicLTI.xml I_514a4081-a0b1-374b-8505-fa2eea9810e0_1_R/BasicLTI.xml I_5150cb7b-788a-3e14-beea-fbfa5a8de75b_1_R/BasicLTI.xml I_516599d1-dd8d-3fda-8aac-d5fb968ae4da_1_R/BasicLTI.xml I_516803a4-998c-375c-ac47-d72e35f50f0f_1_R/BasicLTI.xml I_518692ea-fbe0-3874-8480-0787f8c385d0_R/BasicLTI.xml I_51943444-6ed0-3c1d-b669-40243093514f_1_R/BasicLTI.xml I_51956b6c-7104-3257-99d6-3ac512faf5ae_1_R/BasicLTI.xml I_51b972c5-ef90-327d-88d2-c259408ea78a_R/BasicLTI.xml I_51dccf4a-78a6-32c1-b8ee-dad5a23647de_1_R/BasicLTI.xml I_51fb278f-4bc9-3a24-933e-e925d1b65788_1_R/BasicLTI.xml I_520e2f61-172e-3db3-a359-9329f527dca0_1_R/BasicLTI.xml I_520f8956-996c-368f-9321-8904edfc6f4d_1_R/BasicLTI.xml I_5217d841-82cf-32b7-b1f0-ea65456c710c_1_R/BasicLTI.xml I_5217d841-82cf-32b7-b1f0-ea65456c710c_3_R/BasicLTI.xml I_5217d841-82cf-32b7-b1f0-ea65456c710c_5_R/BasicLTI.xml I_5217d841-82cf-32b7-b1f0-ea65456c710c_7_R/BasicLTI.xml I_521b0d29-0224-3dbf-9c37-5079db05f8f4_R/BasicLTI.xml I_52306e3c-608c-3cda-83fc-f1210e0de955_R/BasicLTI.xml I_5238df15-d3db-3aaf-a0f2-a965d4dcac53_R/BasicLTI.xml I_52573a97-89f5-33c2-8229-ca87f3aea6d6_R/BasicLTI.xml I_5262e57e-6de5-3bb4-ae48-1c09d9ebc282_1_R/BasicLTI.xml I_5271089a-10e4-3eb1-811b-f9e0abf8dd0a_1_R/BasicLTI.xml I_527e93da-3dc6-3ffa-adab-99c3ddaebc3c_1_R/BasicLTI.xml I_528d0a8d-23c7-30d0-bbeb-7a81d0f5df79_R/BasicLTI.xml I_52b0993d-cd1c-3953-bb82-6dce432af28c_1_R/BasicLTI.xml I_52b504be-e8ed-3247-96a5-b7d6750668bc_R/BasicLTI.xml I_52d11045-10bd-3d2b-b880-f6bcffba7770_1_R/BasicLTI.xml I_52d5556f-2526-3813-944e-fbe4c14d7384_1_R/BasicLTI.xml I_52eace85-dea7-3f81-8f52-bc874675e71e_R/BasicLTI.xml I_52f5a19e-af90-3eab-91b2-d28b5ed60110_1_R/BasicLTI.xml I_52f68b1d-6311-3230-bf5d-17b7263d7501_1_R/BasicLTI.xml I_52f9c87d-f9d5-3451-a03f-711d0f4b1cf4_R/BasicLTI.xml I_5323b589-52f2-3623-98c5-3d9a4f50734a_R/BasicLTI.xml I_53277303-5c46-361d-a45b-dbfd54485657_R/BasicLTI.xml I_5337876e-fadf-34e4-a8d6-b837d3fd48dd_1_R/BasicLTI.xml I_53501224-a229-39a5-b1fd-33442dbc8343_R/BasicLTI.xml I_538391a2-ff00-361c-83a5-3a7658019551_1_R/BasicLTI.xml I_53987501-235b-382d-89aa-886d3529881c_1_R/BasicLTI.xml I_53987501-235b-382d-89aa-886d3529881c_3_R/BasicLTI.xml I_53a076ec-915d-3cbd-8afe-bbe32048c468_R/BasicLTI.xml I_53d5265a-68b8-3207-98c0-f153ac18a5bd_1_R/BasicLTI.xml I_53d5265a-68b8-3207-98c0-f153ac18a5bd_3_R/BasicLTI.xml I_53d5265a-68b8-3207-98c0-f153ac18a5bd_5_R/BasicLTI.xml I_53d5265a-68b8-3207-98c0-f153ac18a5bd_7_R/BasicLTI.xml I_53e469b4-f641-3423-a256-c54df4101ebe_R/BasicLTI.xml I_53e99437-72af-395e-8606-13b17e36338c_1_R/BasicLTI.xml I_53fd9f31-1a35-3271-8fd5-509c3b0a24be_1_R/BasicLTI.xml I_53fe4238-8348-3322-b350-9f5ff19dc8e8_R/BasicLTI.xml I_5416dbb1-ab64-37e0-84b0-e55ad291f863_1_R/BasicLTI.xml I_541f3907-8e40-3066-9d39-c1e19e421f5b_R/BasicLTI.xml I_5421f6e2-8ed8-3442-8027-7a34f9f0bdba_R/BasicLTI.xml I_54399393-b910-34ec-8038-83a46e6315d9_1_R/BasicLTI.xml I_543acae1-ca45-3d70-83da-88e67e2cad95_R/BasicLTI.xml I_543f692b-85c7-34ed-a9ef-608fc08dbac9_R/BasicLTI.xml I_5467b058-7c4a-3e57-8b44-b969920512d4_1_R/BasicLTI.xml I_546d1604-6e85-3b9f-9e0a-8aee7300af6e_R/BasicLTI.xml I_54905c25-7e7c-3655-a8f3-8a9e5fbba883_1_R/BasicLTI.xml I_54a4f43e-4437-385a-8a28-62a6c718c800_1_R/BasicLTI.xml I_54a4f43e-4437-385a-8a28-62a6c718c800_3_R/BasicLTI.xml I_54a4f43e-4437-385a-8a28-62a6c718c800_5_R/BasicLTI.xml I_54a4f43e-4437-385a-8a28-62a6c718c800_7_R/BasicLTI.xml I_54bab469-6b18-388a-a45f-dfbe9a6baf67_R/BasicLTI.xml I_54bb0891-bc3b-3100-a5e1-3c27c4472571_1_R/BasicLTI.xml I_54c26e5e-26c0-3e59-9ffb-870ce434423b_1_R/BasicLTI.xml I_54c5630b-c9a8-3bfd-a616-f7e456b30337_1_R/BasicLTI.xml I_54c7ae73-78a9-3910-a7ce-181875b0cb3a_1_R/BasicLTI.xml I_54cb2ce8-32d6-3e01-8c1a-f19d79453402_1_R/BasicLTI.xml I_54d42c89-4627-3596-b06e-6ef2cc63b06a_1_R/BasicLTI.xml I_54d6e9f2-0777-37e5-b350-817f61ddc312_1_R/BasicLTI.xml I_54ddd8c1-0003-3faa-b41c-06b07d6c9d13_1_R/BasicLTI.xml I_54e6b8c9-e662-31bc-8e02-d38e95f7c595_1_R/BasicLTI.xml I_54f6d245-a83f-3cd3-bd39-03e8c0bb522d_1_R/BasicLTI.xml I_54fa6a3b-3f98-3a9d-8816-0ade002c59a7_1_R/BasicLTI.xml I_550c1519-75e9-35e8-8602-cb8974c55efa_1_R/BasicLTI.xml I_55118ee6-2086-3d18-b267-cac78f33e6fb_1_R/BasicLTI.xml I_5521e419-5505-3b3d-a533-eec4f4e476e8_1_R/BasicLTI.xml I_5524619e-6f7f-31e3-8ab8-3aa890bf1fa0_1_R/BasicLTI.xml I_55418b7a-b02c-3065-a726-da37c4419fc2_R/BasicLTI.xml I_554807af-8e85-3634-8f2e-ae12c6ea102b_1_R/BasicLTI.xml I_5583b738-07b6-3ae6-b110-a07530f75bfe_R/BasicLTI.xml I_55840f3e-5787-3673-a42e-a06736118d11_1_R/BasicLTI.xml I_559d0797-4609-3b81-9232-23817800aa95_1_R/BasicLTI.xml I_559e0c77-4086-3e9b-b459-0b57a61e2c16_1_R/BasicLTI.xml I_55c2d0c3-2d20-3a54-ac21-f92b0296362f_1_R/BasicLTI.xml I_55c541ec-7773-3eff-9d94-60e0ac14057c_R/BasicLTI.xml I_55cedbc8-97e6-30bc-bbd9-bb04942e9744_R/BasicLTI.xml I_55e754e3-0868-3f3b-bffb-4b32bca539df_R/BasicLTI.xml I_5616c2d4-8a50-36a6-aa01-216f77406633_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_11_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_13_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_15_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_1_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_3_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_5_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_7_R/BasicLTI.xml I_56191e2b-d469-38f8-a19f-201cf74d69f4_9_R/BasicLTI.xml I_5645a048-3baa-39cb-a1ae-e0c4b5922b49_R/BasicLTI.xml I_565500e2-e90f-3f07-9aaa-00c69e028c0a_1_R/BasicLTI.xml I_56572a5c-068c-3b83-b924-e1ef076e10cf_R/BasicLTI.xml I_567a2880-16d6-3137-b0c3-75e4d4b803e9_1_R/BasicLTI.xml I_567a2880-16d6-3137-b0c3-75e4d4b803e9_3_R/BasicLTI.xml I_567ca80d-fc18-3201-932d-5bb12614cb26_1_R/BasicLTI.xml I_567ca80d-fc18-3201-932d-5bb12614cb26_3_R/BasicLTI.xml I_568524c3-c1d1-3d34-bbeb-b5872ad30b12_1_R/BasicLTI.xml I_568524c3-c1d1-3d34-bbeb-b5872ad30b12_3_R/BasicLTI.xml I_56ad25e8-a8c1-3834-9498-8a37619cfea7_1_R/BasicLTI.xml I_56d16349-c348-3ebf-825f-c027315dcfe1_R/BasicLTI.xml I_56dd5938-837b-315c-99f3-5fe2d7b21483_1_R/BasicLTI.xml I_56e06932-2aed-317f-ac28-a511ccaecf6c_R/BasicLTI.xml I_56e92269-d480-3d6c-8062-598884ec7a42_1_R/BasicLTI.xml I_56ea5e0b-a546-32f0-9209-eeade2b75662_R/BasicLTI.xml I_56eb0bdb-077f-3078-9b79-cf1fdf2338e7_1_R/BasicLTI.xml I_56f66051-e349-396b-aee4-e48fe1cd3c46_1_R/BasicLTI.xml I_56fc09bb-5231-3b81-af8c-52ca76349434_R/BasicLTI.xml I_57039874-694b-3e87-bcf5-986d611be16b_1_R/BasicLTI.xml I_5708656b-f7ee-3846-b94d-5d42eef64047_1_R/BasicLTI.xml I_5713b28f-b8de-369f-9e74-ce7ce449ec85_1_R/BasicLTI.xml I_5745f1d5-cd1d-3059-b037-fde48c15a51f_R/BasicLTI.xml I_575d31a7-3277-3285-8855-dbf4101e5647_1_R/BasicLTI.xml I_576f35e1-9b3d-3d2f-838c-0643279498f7_1_R/BasicLTI.xml I_5772a91a-46ab-3d6d-807e-b3261f9a03ca_1_R/BasicLTI.xml I_577c5bd5-b88d-3616-b97c-8a0f0cfba694_1_R/BasicLTI.xml I_57aa8036-46b3-39b0-854f-d9ab446cbbdd_R/BasicLTI.xml I_57b0bf8b-3fbd-3132-82ba-e064660d1180_1_R/BasicLTI.xml I_57b7ff48-f542-3bb3-bf22-a539de902237_R/BasicLTI.xml I_57ddd196-9553-37d0-844f-7697ad3eb03a_1_R/BasicLTI.xml I_57e20f99-d881-38e1-91c8-0d62ea8d969f_1_R/BasicLTI.xml I_57f57e33-1a60-3367-98c3-300e5c02b5b6_1_R/BasicLTI.xml I_57f69e88-42cc-392e-be00-119fc00e45d2_R/BasicLTI.xml I_57f737a8-42f8-3c48-aaa3-8577a96833db_R/BasicLTI.xml I_57f7eaa1-7e48-3ce1-9835-9e61f2d642ee_1_R/BasicLTI.xml I_580e920a-d925-3541-9da9-e69ad61405a9_1_R/BasicLTI.xml I_5815dfce-c471-3d4f-864a-0ff7b333bc8e_1_R/BasicLTI.xml I_58180aa3-db32-3340-8a80-f8f2c951cb8a_R/BasicLTI.xml I_582bd575-92e8-3887-8f99-d24920afcb3c_R/BasicLTI.xml I_58333c03-948f-3120-8b63-a0cae0dcc777_1_R/BasicLTI.xml I_5844b145-43d8-3c7c-84c1-a06ab6015ead_1_R/BasicLTI.xml I_586422d8-80e3-30b1-a572-714b64f506fa_R/BasicLTI.xml I_586bbaeb-c398-3510-aaef-d16ad3a1c3d4_R/BasicLTI.xml I_586c7645-9b3f-3a7d-941a-bd426eaa9dbe_1_R/BasicLTI.xml I_5885df05-adae-37eb-9639-dfd0e9eded4a_R/BasicLTI.xml I_589b2bc0-a519-319e-b849-ff0599b7c61e_1_R/BasicLTI.xml I_589f9979-e491-3028-8cab-644ade39e456_R/BasicLTI.xml I_58a07340-263f-3925-8541-4fc0efe783f8_1_R/BasicLTI.xml I_58a07340-263f-3925-8541-4fc0efe783f8_3_R/BasicLTI.xml I_58ac14eb-3072-3ca3-a73f-f94af4d9f5e4_1_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_11_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_1_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_3_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_5_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_7_R/BasicLTI.xml I_58cc4861-ddeb-35cc-af85-0692055882b7_9_R/BasicLTI.xml I_58f1e239-59ce-349c-8170-605617644f11_1_R/BasicLTI.xml I_5905316b-b517-3b8f-ba83-d5ad129b51e7_R/BasicLTI.xml I_59203846-8e0a-3f7d-ac81-5e48ab528b87_1_R/BasicLTI.xml I_59203846-8e0a-3f7d-ac81-5e48ab528b87_3_R/BasicLTI.xml I_59203846-8e0a-3f7d-ac81-5e48ab528b87_5_R/BasicLTI.xml I_59203846-8e0a-3f7d-ac81-5e48ab528b87_7_R/BasicLTI.xml I_59203846-8e0a-3f7d-ac81-5e48ab528b87_9_R/BasicLTI.xml I_59228da0-f568-306c-a1f4-6e3f965ce51c_1_R/BasicLTI.xml I_59228da0-f568-306c-a1f4-6e3f965ce51c_3_R/BasicLTI.xml I_59228da0-f568-306c-a1f4-6e3f965ce51c_5_R/BasicLTI.xml I_59244b95-4503-37d8-9b61-b5b37ea7c168_1_R/BasicLTI.xml I_59362a27-2ce3-3011-9cf3-2ffbfbfc96eb_1_R/BasicLTI.xml I_593fcac7-ee2e-313f-97ac-926d3258fadd_1_R/BasicLTI.xml I_594c6e43-4d76-32aa-8847-bcf29bf4992f_R/BasicLTI.xml I_595624b6-bb95-3bc2-b95a-371364a6d9bd_1_R/BasicLTI.xml I_595624b6-bb95-3bc2-b95a-371364a6d9bd_3_R/BasicLTI.xml I_5973018c-2e98-3f5f-9c10-d3d04e7e41a9_1_R/BasicLTI.xml I_5973018c-2e98-3f5f-9c10-d3d04e7e41a9_3_R/BasicLTI.xml I_597aa2df-5173-3f98-b47d-07f3cc02f0e6_R/BasicLTI.xml I_59ca3cf7-b631-319f-83d2-c727f1205763_R/BasicLTI.xml I_59cf7001-c395-3b1e-bd7b-90284facbb5b_1_R/BasicLTI.xml I_59d6cea0-3088-319a-b726-ed903edeadab_R/BasicLTI.xml I_5a001589-a0d4-39c5-b600-cf54ba4d2287_1_R/BasicLTI.xml I_5a001589-a0d4-39c5-b600-cf54ba4d2287_3_R/BasicLTI.xml I_5a0cff74-303a-300c-a9a5-a7b776c05bca_1_R/BasicLTI.xml I_5a40276d-26b3-3b8e-b9da-a44b7aa836f8_1_R/BasicLTI.xml I_5a444f7f-9c91-31e4-be9b-23ebef5d14a5_1_R/BasicLTI.xml I_5a5d0056-3adb-3b18-8826-40c5beb6624f_1_R/BasicLTI.xml I_5a87110c-95f2-3225-b774-e7d33671a58e_1_R/BasicLTI.xml I_5ab26066-51ed-381c-92f8-bf4ead96ef4e_1_R/BasicLTI.xml I_5ad92ddb-6869-3c5c-afbe-555efbcccf18_1_R/BasicLTI.xml I_5ae3db81-b5e6-3671-923d-8caf053bb6dd_1_R/BasicLTI.xml I_5afb1408-f89c-3b66-918b-422262ffb72c_R/BasicLTI.xml I_5afc29ac-428f-30ca-9b25-03152a701fe9_1_R/BasicLTI.xml I_5afc29ac-428f-30ca-9b25-03152a701fe9_3_R/BasicLTI.xml I_5afc29ac-428f-30ca-9b25-03152a701fe9_5_R/BasicLTI.xml I_5afc29ac-428f-30ca-9b25-03152a701fe9_7_R/BasicLTI.xml I_5b0e7acd-99f6-37ee-96c3-a03df6fc9f7c_1_R/BasicLTI.xml I_5b0fd2b8-01ac-3063-948b-6a288ba598a4_1_R/BasicLTI.xml I_5b12f8c1-e5fa-30f7-8990-efeb41dc71ef_R/BasicLTI.xml I_5b136302-1f19-37b8-85cb-083b045ce14c_1_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_11_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_13_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_1_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_3_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_5_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_7_R/BasicLTI.xml I_5b13d6b1-6459-3c57-8600-abdc2f9a3aaf_9_R/BasicLTI.xml I_5b1793fb-a580-3181-a481-701a321f95b7_1_R/BasicLTI.xml I_5b3dcc15-1e70-3e5a-ae97-c0aab31ee11d_1_R/BasicLTI.xml I_5b3dcc15-1e70-3e5a-ae97-c0aab31ee11d_3_R/BasicLTI.xml I_5b3dcc15-1e70-3e5a-ae97-c0aab31ee11d_5_R/BasicLTI.xml I_5b3dcc15-1e70-3e5a-ae97-c0aab31ee11d_7_R/BasicLTI.xml I_5b3f35ce-8213-3fab-9ef8-e9b83a90fa6b_1_R/BasicLTI.xml I_5b46a3bc-679b-32aa-bdf0-42018194fea6_1_R/BasicLTI.xml I_5b6806d6-07a5-37ce-9974-1b467fe2a740_1_R/BasicLTI.xml I_5b6a4dc8-a00f-3329-924f-68fc4cb53e86_1_R/BasicLTI.xml I_5b7585d7-293e-3536-b728-9243e5c81b49_R/BasicLTI.xml I_5bcf1bcb-7ff0-31e0-ac88-b2d391701d28_1_R/BasicLTI.xml I_5bcf1bcb-7ff0-31e0-ac88-b2d391701d28_3_R/BasicLTI.xml I_5bcf1bcb-7ff0-31e0-ac88-b2d391701d28_5_R/BasicLTI.xml I_5bf40443-b505-3ff9-a9da-ecf3b398718c_1_R/BasicLTI.xml I_5bfa862c-0512-3c58-8130-2dc110c5a67b_R/BasicLTI.xml I_5c0019eb-b854-3eec-8dff-5d3035037593_1_R/BasicLTI.xml I_5c501929-6f8a-379b-a256-47cfe5dceb9b_1_R/BasicLTI.xml I_5c8f1ba1-f895-34fa-ac82-cb66437bb304_R/BasicLTI.xml I_5c97b1bb-4227-3e51-86eb-0a2c6093dfc2_1_R/BasicLTI.xml I_5c97b1bb-4227-3e51-86eb-0a2c6093dfc2_3_R/BasicLTI.xml I_5c989fd9-8bfc-3573-b0f4-0164ea51f484_1_R/BasicLTI.xml I_5c989fd9-8bfc-3573-b0f4-0164ea51f484_3_R/BasicLTI.xml I_5c989fd9-8bfc-3573-b0f4-0164ea51f484_5_R/BasicLTI.xml I_5c989fd9-8bfc-3573-b0f4-0164ea51f484_7_R/BasicLTI.xml I_5c9f61ed-5608-32d7-a650-b0fba6ebd31f_R/BasicLTI.xml I_5cbbe1fb-8d5f-3117-8b23-e03f052c9fda_R/BasicLTI.xml I_5cbcc403-4ab4-34cb-a903-24200f3078c1_1_R/BasicLTI.xml I_5cc5048e-3721-3bf7-96ee-78b34cfc09c1_R/BasicLTI.xml I_5ccf0253-a998-3d6c-80b0-f4df9c4cf1f1_R/BasicLTI.xml I_5cdc2cfa-d5c4-3e86-ac1a-b720aac9fae3_1_R/BasicLTI.xml I_5d147ba8-2ef2-3b48-92d3-4ced0c65c0f0_1_R/BasicLTI.xml I_5d151216-7557-3bc5-9b63-13e5bd495fdf_R/BasicLTI.xml I_5d3c8c28-9500-3369-9bcd-9db34a574abf_1_R/BasicLTI.xml I_5d574e93-de30-3088-bf01-7e1e26eebcd8_R/BasicLTI.xml I_5d5cccca-51ca-3646-993a-65fea52c88c5_R/BasicLTI.xml I_5d65c13f-25d6-372a-835c-e47390b020c0_1_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_11_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_1_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_3_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_5_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_7_R/BasicLTI.xml I_5d71e079-cd50-3e15-ab90-3b634a588c2e_9_R/BasicLTI.xml I_5d758bb1-3ce4-3d76-8c49-d942d766e8da_1_R/BasicLTI.xml I_5d77e813-f64c-314b-9586-ad32ec2eb178_1_R/BasicLTI.xml I_5d8169f7-02a7-33aa-90d8-58f6b21a81b1_1_R/BasicLTI.xml I_5d881b82-cbc1-39a7-801f-ec071fa50619_1_R/BasicLTI.xml I_5da7772c-c4c9-360f-acf4-40de6c28542d_1_R/BasicLTI.xml I_5daa1a9a-2443-3053-9263-fa2b28dccffa_R/BasicLTI.xml I_5dbe90d5-04cc-346d-82f8-24b0090be9c2_1_R/BasicLTI.xml I_5dc3082a-f8b7-381c-a391-22555e59045c_R/BasicLTI.xml I_5dd9e5a9-d4a1-3d74-ba9a-03c2f5b711ad_1_R/BasicLTI.xml I_5dd9e5a9-d4a1-3d74-ba9a-03c2f5b711ad_3_R/BasicLTI.xml I_5df50ffa-f2b9-3e14-9382-7d3e0705aa23_R/BasicLTI.xml I_5df9f690-ae30-345e-843b-dd81e91b12f0_1_R/BasicLTI.xml I_5df9f690-ae30-345e-843b-dd81e91b12f0_3_R/BasicLTI.xml I_5df9f690-ae30-345e-843b-dd81e91b12f0_5_R/BasicLTI.xml I_5df9f690-ae30-345e-843b-dd81e91b12f0_7_R/BasicLTI.xml I_5e0582fe-b6ad-3a18-b1e0-bd5fb645e378_1_R/BasicLTI.xml I_5e2abe11-124c-358a-8644-ec829535d5a0_1_R/BasicLTI.xml I_5e5047a3-caf5-3656-8064-f34757cf4304_1_R/BasicLTI.xml I_5e5047a3-caf5-3656-8064-f34757cf4304_3_R/BasicLTI.xml I_5e5047a3-caf5-3656-8064-f34757cf4304_5_R/BasicLTI.xml I_5e5047a3-caf5-3656-8064-f34757cf4304_7_R/BasicLTI.xml I_5e6b5342-594a-378b-96db-46b2dd5a8145_1_R/BasicLTI.xml I_5e6c99c2-557d-37de-97af-7026f12c79e6_1_R/BasicLTI.xml I_5e6c99c2-557d-37de-97af-7026f12c79e6_3_R/BasicLTI.xml I_5e915ee9-02b3-3ecb-80b2-cdf9b0b7496d_1_R/BasicLTI.xml I_5eaba0d5-57c9-37ce-aec2-cdb9d10ba36d_1_R/BasicLTI.xml I_5f0bc24f-0bc5-3615-b008-ad94f8557e8e_R/BasicLTI.xml I_5f1e08b3-4f57-390a-938e-89c2201e4176_1_R/BasicLTI.xml I_5fa6b52c-a502-31df-aca8-59a2f347ac81_R/BasicLTI.xml I_5fa89c56-73fd-388a-a0fb-9dafa8441629_1_R/BasicLTI.xml I_5ffea17e-3473-3cd8-880c-177b41ea7d69_1_R/BasicLTI.xml I_5ffea17e-3473-3cd8-880c-177b41ea7d69_3_R/BasicLTI.xml I_5ffea17e-3473-3cd8-880c-177b41ea7d69_5_R/BasicLTI.xml I_60039673-930a-326a-84d9-8a9ac7374aaf_R/BasicLTI.xml I_602e4d8e-240e-3f29-b19c-9480ee591a8c_R/BasicLTI.xml I_603f2506-2a21-37e1-9395-dfa0d5c10160_R/BasicLTI.xml I_606b2fe5-20b4-3706-87df-72e6bccc6790_1_R/BasicLTI.xml I_6080241f-aa44-3cc7-9b5a-26f8afb9f030_1_R/BasicLTI.xml I_608f1e5b-812e-3acd-a6f1-7b6a37cbdd12_R/BasicLTI.xml I_6096fdc8-53a7-37c1-9982-a4b2ad97634f_1_R/BasicLTI.xml I_60a6cd82-105e-3bb3-b571-886341927858_R/BasicLTI.xml I_60a8bec3-f315-3d35-a5fc-00c1dc631e8e_1_R/BasicLTI.xml I_60bfaf50-d77d-3bd7-aa44-38e03892667b_1_R/BasicLTI.xml I_60c29bd5-1744-3295-8587-029061272d8c_R/BasicLTI.xml I_60e60315-02c7-3296-b9a0-533fb310fb33_1_R/BasicLTI.xml I_60e60315-02c7-3296-b9a0-533fb310fb33_3_R/BasicLTI.xml I_60e9c8a6-09b6-318e-8f25-36ae0b4f852a_1_R/BasicLTI.xml I_61113ee7-5104-3496-a35b-a3ceaf1869ad_1_R/BasicLTI.xml I_611d8b1b-619a-385e-95ef-3d77cdb29574_1_R/BasicLTI.xml I_6144ecbe-59c9-3927-9e6c-4b9ff55711b4_R/BasicLTI.xml I_614d4f50-c566-3e81-a17d-be59e74c060d_1_R/BasicLTI.xml I_614d4f50-c566-3e81-a17d-be59e74c060d_3_R/BasicLTI.xml I_614d4f50-c566-3e81-a17d-be59e74c060d_5_R/BasicLTI.xml I_614d4f50-c566-3e81-a17d-be59e74c060d_7_R/BasicLTI.xml I_614d4f50-c566-3e81-a17d-be59e74c060d_9_R/BasicLTI.xml I_6164a974-863a-30fb-b31f-a9917ed07b52_1_R/BasicLTI.xml I_6171d242-9a21-3d4a-96d2-a23c261f38dd_R/BasicLTI.xml I_61801b77-3e91-3559-b962-3e2aa0148b0d_1_R/BasicLTI.xml I_6181f954-9df7-3024-9572-8941b19aff6a_1_R/BasicLTI.xml I_61a33fca-3fac-31c9-88ca-03a20a26323b_1_R/BasicLTI.xml I_61c6208b-67ce-3412-98c7-fa19cf03db00_R/BasicLTI.xml I_61cdbd3e-10d4-3cee-b00e-ade44af7a5b9_R/BasicLTI.xml I_61d735ac-7e18-3d1c-989d-f4f7f5b8712c_R/BasicLTI.xml I_61db4a67-a8c5-3629-81fb-cef4d4dfe944_1_R/BasicLTI.xml I_61e121b5-962b-375e-97b0-a34b81ce9164_1_R/BasicLTI.xml I_61e3696f-e7ad-36b5-a3c2-e5f7241e4018_1_R/BasicLTI.xml I_61f2db9a-fb88-3a58-afbe-e0e3afc307c4_1_R/BasicLTI.xml I_61f2db9a-fb88-3a58-afbe-e0e3afc307c4_3_R/BasicLTI.xml I_61fef315-3c8f-3299-87bf-50fb55e10952_1_R/BasicLTI.xml I_620949b7-9c76-3ac5-b42b-4792ff19e3d9_R/BasicLTI.xml I_620d1b30-a624-39a7-b794-7301f61da558_1_R/BasicLTI.xml I_621a007b-96e8-3855-a5c4-d0efb06d16d1_1_R/BasicLTI.xml I_6229c640-b784-3817-93a5-0a47ec7bebdb_1_R/BasicLTI.xml I_6229c640-b784-3817-93a5-0a47ec7bebdb_3_R/BasicLTI.xml I_6229c640-b784-3817-93a5-0a47ec7bebdb_5_R/BasicLTI.xml I_624a0544-72b4-3866-a9f2-ede3a8c584f3_1_R/BasicLTI.xml I_6251e683-3822-334d-a4b0-b5e92e861147_1_R/BasicLTI.xml I_625b8280-f8c0-31d6-a46a-cf42b69bf902_R/BasicLTI.xml I_626b22a8-861f-3cbb-8caf-92aaaefeb1a7_1_R/BasicLTI.xml I_626e5160-d061-3a00-9bb9-b5f209f9b6c9_1_R/BasicLTI.xml I_6282819f-4ef4-310f-83d9-6725d0d1d8ea_1_R/BasicLTI.xml I_628c54dd-1f8e-333e-a5f2-963749a6a5a2_1_R/BasicLTI.xml I_629e4959-459c-34f1-b730-6cd5dc1f48e9_R/BasicLTI.xml I_62cf3f79-c4d8-3dc4-8324-b7dabe10fa19_R/BasicLTI.xml I_62d09b07-4a1b-3c21-b237-8826c17443f8_1_R/BasicLTI.xml I_62ef1be4-eac8-325c-bc5d-96af1d82522d_1_R/BasicLTI.xml I_62fca85f-6039-3863-80e6-ba7fe0873948_1_R/BasicLTI.xml I_6316aad1-c813-305d-9cdf-a6495e1c3a73_R/BasicLTI.xml I_6318dc72-6157-3cf7-aa7c-b8e060e1010a_1_R/BasicLTI.xml I_631d7a9e-96e5-3f1c-b86e-c02d17125501_1_R/BasicLTI.xml I_631d7a9e-96e5-3f1c-b86e-c02d17125501_3_R/BasicLTI.xml I_631d7a9e-96e5-3f1c-b86e-c02d17125501_5_R/BasicLTI.xml I_631d7a9e-96e5-3f1c-b86e-c02d17125501_7_R/BasicLTI.xml I_631d7a9e-96e5-3f1c-b86e-c02d17125501_9_R/BasicLTI.xml I_6346827a-880c-34d2-b36c-052e8db374f6_1_R/BasicLTI.xml I_6348fd2a-a5d2-3c88-9e4f-7cc538325ddf_R/BasicLTI.xml I_635231bb-d7c7-3eeb-a98d-a36a2238e2fb_R/BasicLTI.xml I_6393d075-eb89-3f56-a50d-9744340afcc2_R/BasicLTI.xml I_63970b35-143f-34fa-b22e-06a310fcc829_1_R/BasicLTI.xml I_63a69e8a-f751-36c5-b7e3-8fc0f473fbab_1_R/BasicLTI.xml I_63ba803a-25ac-393e-bdb8-eded325c806a_1_R/BasicLTI.xml I_63bafff1-8234-3cf6-b193-581dbe54e0ae_1_R/BasicLTI.xml I_63cba4d0-f290-3db1-b186-88812c5d9bff_1_R/BasicLTI.xml I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_1_R/BasicLTI.xml I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_3_R/BasicLTI.xml I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_5_R/BasicLTI.xml I_63f4a00f-cb3a-34a1-b225-ba1b37f94271_7_R/BasicLTI.xml I_640577b4-a858-32cd-9630-1e0ed4603df3_1_R/BasicLTI.xml I_64475f0b-1ee4-370d-827c-842dba53904a_1_R/BasicLTI.xml I_645ccefc-923a-396a-8294-27415d4d7a50_1_R/BasicLTI.xml I_64651953-c1b7-35c8-9240-9201c335f335_R/BasicLTI.xml I_646e1cac-56f7-3a1e-8ed9-30a163c9bb8f_1_R/BasicLTI.xml I_6483756b-a335-3ccb-899e-21b8fde001c9_R/BasicLTI.xml I_648f9b26-0444-3db5-b009-270ce36267f4_1_R/BasicLTI.xml I_649187ee-33e3-3f05-8e35-72f406ff3d86_1_R/BasicLTI.xml I_64ad9561-e7dd-3b2b-8cd1-19b83aaf0edf_1_R/BasicLTI.xml I_64b9cbca-de0e-3c8b-a2ec-0e92f62fb547_1_R/BasicLTI.xml I_64be3d81-10df-3618-9f46-cd567d7c0182_R/BasicLTI.xml I_64ee53b1-b510-3d41-b4f8-3272b26c6a8d_R/BasicLTI.xml I_64f54a94-06ce-3da4-bd1f-ac4aafc9a441_1_R/BasicLTI.xml I_651d72a9-076b-3ed4-955a-0337d2bc76f6_R/BasicLTI.xml I_653121b2-0891-36ea-9a99-111f3652c31d_R/BasicLTI.xml I_6534bd62-cbe8-3e3c-9974-9235fe1b4db1_R/BasicLTI.xml I_653bdc98-c8e9-349e-94c0-2a5bc204b864_1_R/BasicLTI.xml I_6559d7e3-6fca-3c60-ab20-99c1a2826dbf_R/BasicLTI.xml I_659bdb07-2095-368f-802f-2caa5e70c3f7_1_R/BasicLTI.xml I_65b8d8e3-f1f8-37e0-9545-f8824e300b2d_1_R/BasicLTI.xml I_65f1ef1e-a3b2-3b54-b8e9-c6c4a97a3949_1_R/BasicLTI.xml I_6624b392-5ed8-336c-98a6-42d906a64922_1_R/BasicLTI.xml I_6624b392-5ed8-336c-98a6-42d906a64922_3_R/BasicLTI.xml I_6638b296-78e1-3a83-a6c3-2c432d7b2085_R/BasicLTI.xml I_66714445-8996-358a-8b9c-25bc938d25b7_1_R/BasicLTI.xml I_66714445-8996-358a-8b9c-25bc938d25b7_3_R/BasicLTI.xml I_667cda6c-ae6a-3e32-82d2-4a523d0b0184_R/BasicLTI.xml I_667e1cb7-c64c-3e3c-87c0-de33df801e67_1_R/BasicLTI.xml I_668be9c9-352a-304c-8c2c-36de679c433a_1_R/BasicLTI.xml I_66b1a834-8ccb-323d-a040-3877f6d6854a_1_R/BasicLTI.xml I_66c56d1d-bec9-326b-9c35-3b71f47feea7_1_R/BasicLTI.xml I_66cb1617-21dc-3aae-a873-aeae7fe59182_1_R/BasicLTI.xml I_66e42faf-a5a1-3c5e-b497-1303aaca83b9_1_R/BasicLTI.xml I_66e8af22-a544-3a7b-a3de-6f46071776ac_R/BasicLTI.xml I_66f75c32-8348-3506-b175-50ce4182f6ee_R/BasicLTI.xml I_66fd7bc2-bdf6-32a1-a34d-d6ab56fa7c8c_1_R/BasicLTI.xml I_66fd7bc2-bdf6-32a1-a34d-d6ab56fa7c8c_3_R/BasicLTI.xml I_66fd7bc2-bdf6-32a1-a34d-d6ab56fa7c8c_5_R/BasicLTI.xml I_66fd7bc2-bdf6-32a1-a34d-d6ab56fa7c8c_7_R/BasicLTI.xml I_67007dfb-58ee-3011-b5a3-d9e24a89d4d9_R/BasicLTI.xml I_6701b029-dfe5-398d-9b62-de47b362c9ec_R/BasicLTI.xml I_671f29a4-4e1c-3482-9689-3fed2313f452_R/BasicLTI.xml I_672e5ecd-231d-31de-a8d4-9aae18e6ee27_1_R/BasicLTI.xml I_672e5ecd-231d-31de-a8d4-9aae18e6ee27_3_R/BasicLTI.xml I_676a2e20-8405-305c-a0bb-98cfafd74318_1_R/BasicLTI.xml I_676a2e20-8405-305c-a0bb-98cfafd74318_3_R/BasicLTI.xml I_677750eb-9ba2-3d63-a174-5c21f65a091b_1_R/BasicLTI.xml I_67814478-17c7-3df0-b9dc-915e668aded7_1_R/BasicLTI.xml I_6785e65a-f99e-3e6c-9433-c0b20e97b90f_1_R/BasicLTI.xml I_678aa654-103c-3872-b510-d42d92bfc021_1_R/BasicLTI.xml I_67a043fa-d11b-330f-a36f-d3fc49c2293f_R/BasicLTI.xml I_67afc408-3a40-3f4c-b876-8118e35a5018_1_R/BasicLTI.xml I_67afc408-3a40-3f4c-b876-8118e35a5018_3_R/BasicLTI.xml I_67afc408-3a40-3f4c-b876-8118e35a5018_5_R/BasicLTI.xml I_67afc408-3a40-3f4c-b876-8118e35a5018_7_R/BasicLTI.xml I_67cb7069-066d-3d22-9f2e-0ed6b0b11670_1_R/BasicLTI.xml I_67cd9260-438e-3204-9253-78dedbf74186_1_R/BasicLTI.xml I_67cd9260-438e-3204-9253-78dedbf74186_3_R/BasicLTI.xml I_67cd9260-438e-3204-9253-78dedbf74186_5_R/BasicLTI.xml I_67fe2ace-055b-36c2-8d57-d6123e2dbd85_1_R/BasicLTI.xml I_67fe499d-9a48-30d1-8dc8-ae1d12aca234_1_R/BasicLTI.xml I_6820d458-4413-357f-938d-859acc4bb02d_R/BasicLTI.xml I_68216a61-9662-32ff-965e-936bdd64f93d_1_R/BasicLTI.xml I_68220337-1552-3c5f-9546-83bc24af7dab_R/BasicLTI.xml I_682dd6fd-c0fc-3f8d-827d-c8743103615b_1_R/BasicLTI.xml I_6864721f-3d2a-349c-8ed9-f65746104f40_R/BasicLTI.xml I_687b3b14-7e4f-3cae-be55-ee11b37e50a3_R/BasicLTI.xml I_68807928-7884-3dcc-8882-ee28667245ec_R/BasicLTI.xml I_68b096ca-9bc3-3acb-b06c-38d3368bd596_1_R/BasicLTI.xml I_68d815dd-fa1a-304d-a74d-4ae43aa22dad_R/BasicLTI.xml I_690140c1-24fa-3b42-8de5-b515a6e1aa91_1_R/BasicLTI.xml I_690140c1-24fa-3b42-8de5-b515a6e1aa91_3_R/BasicLTI.xml I_690140c1-24fa-3b42-8de5-b515a6e1aa91_5_R/BasicLTI.xml I_690140c1-24fa-3b42-8de5-b515a6e1aa91_7_R/BasicLTI.xml I_690140c1-24fa-3b42-8de5-b515a6e1aa91_9_R/BasicLTI.xml I_690eeb39-6876-33a0-aadb-24447c506207_1_R/BasicLTI.xml I_6912f1f2-1be0-3e49-b1e7-fab25b5a726e_1_R/BasicLTI.xml I_691571ed-aeaa-3749-8ee2-fd467d8358ef_R/BasicLTI.xml I_6922642f-1219-31f6-9a84-3edd6e51979d_R/BasicLTI.xml I_6984f282-b3c4-3b68-8c08-5e347464d9da_1_R/BasicLTI.xml I_6984f282-b3c4-3b68-8c08-5e347464d9da_3_R/BasicLTI.xml I_69862cc5-89bf-370e-918a-f722d30762c5_1_R/BasicLTI.xml I_699beac3-df6f-3bac-a5c8-8c11cabd06e9_1_R/BasicLTI.xml I_699f1d02-3489-3206-927d-deea20720831_R/BasicLTI.xml I_69a29a97-f533-3097-9dae-78998909d660_R/BasicLTI.xml I_69b5f7ab-483a-3a33-b1f8-f70b933a31d5_1_R/BasicLTI.xml I_69c74adc-b4ef-3438-89f0-7879d700367e_1_R/BasicLTI.xml I_6a14b85a-64ff-378d-94ca-f2b8898d579a_R/BasicLTI.xml I_6a185fa8-e532-341f-b91d-cd4ce7538bf0_R/BasicLTI.xml I_6a319894-0bbc-34fb-8db9-efa016ff21e5_R/BasicLTI.xml I_6a3d58a4-e9bf-3329-97d6-57c0ed437ad6_R/BasicLTI.xml I_6a50bd40-39b8-3c21-941f-c445cc116b18_R/BasicLTI.xml I_6a62d84b-a483-3fcb-8e75-9389b184deb2_1_R/BasicLTI.xml I_6a62d84b-a483-3fcb-8e75-9389b184deb2_3_R/BasicLTI.xml I_6a87f20b-2852-3dd7-b2a6-b56df632ceaa_1_R/BasicLTI.xml I_6ad07724-0666-39ea-a6bb-55e502f5156a_1_R/BasicLTI.xml I_6af577cf-192f-3e87-849f-373e01e030e1_R/BasicLTI.xml I_6affbf30-687d-328c-988e-2c2ef7cb53e7_1_R/BasicLTI.xml I_6b9f3fc4-0d00-354f-8710-0d93b67fcc92_R/BasicLTI.xml I_6badc183-298f-3c32-9d67-9425bc70794c_1_R/BasicLTI.xml I_6bb6e085-43a1-3633-bc5a-56ea36b1433d_1_R/BasicLTI.xml I_6bc54c7d-f0e1-3d84-a7bb-2fde6643e07f_1_R/BasicLTI.xml I_6bd6ff2b-8866-3081-a7c4-54ca3e7a43a9_1_R/BasicLTI.xml I_6bdace42-2db5-3071-9f7c-e2ae6882fdc5_1_R/BasicLTI.xml I_6bdc953f-afd2-3f0a-9c47-8800b3d0ff1c_R/BasicLTI.xml I_6be8e0bb-9f6b-3d56-be62-46115b658f5c_1_R/BasicLTI.xml I_6be8e0bb-9f6b-3d56-be62-46115b658f5c_3_R/BasicLTI.xml I_6c01ebfe-3cea-331d-b392-c5ceecae858a_1_R/BasicLTI.xml I_6c0e4dce-d7d3-3422-8bc7-3784fded9d1e_1_R/BasicLTI.xml I_6c10171a-7bcb-31e9-8448-d96151e14fbc_1_R/BasicLTI.xml I_6c10171a-7bcb-31e9-8448-d96151e14fbc_3_R/BasicLTI.xml I_6c10171a-7bcb-31e9-8448-d96151e14fbc_5_R/BasicLTI.xml I_6c10171a-7bcb-31e9-8448-d96151e14fbc_7_R/BasicLTI.xml I_6c120ace-834f-3828-88ea-e64dc5910348_1_R/BasicLTI.xml I_6c2c1eb6-97e7-3590-aeaa-f18acb4d3515_1_R/BasicLTI.xml I_6c45eecd-8ce8-3457-98b6-ebeb9ef07456_1_R/BasicLTI.xml I_6c4cef6a-f412-3ede-98af-ea485c1df042_R/BasicLTI.xml I_6c4d57d2-c45e-31da-8bcf-e4339f0b924b_1_R/BasicLTI.xml I_6c611f0c-9d5f-3e7b-b67a-afc181431a0c_R/BasicLTI.xml I_6c801630-a9e8-3848-8520-a828b273d316_R/BasicLTI.xml I_6c8e2aba-4c3b-3f29-8cb4-fde1abc6d039_1_R/BasicLTI.xml I_6c8e2aba-4c3b-3f29-8cb4-fde1abc6d039_3_R/BasicLTI.xml I_6c8e2aba-4c3b-3f29-8cb4-fde1abc6d039_5_R/BasicLTI.xml I_6c8e2aba-4c3b-3f29-8cb4-fde1abc6d039_7_R/BasicLTI.xml I_6c8e4952-3578-3899-bcd7-04938eaf8509_R/BasicLTI.xml I_6c8eabe6-8108-3edc-8490-1c453b22dd44_1_R/BasicLTI.xml I_6c8eabe6-8108-3edc-8490-1c453b22dd44_3_R/BasicLTI.xml I_6c9718dc-da95-391f-b01e-8f38d4789099_R/BasicLTI.xml I_6ca09a4b-af4e-30f7-97d8-5e5fd5af43fd_R/BasicLTI.xml I_6cb4012a-ec89-3cf2-a175-f773be18f584_1_R/BasicLTI.xml I_6cbc2764-2315-3a58-a3dd-9c9fa7a712fe_R/BasicLTI.xml I_6cc4929e-57a3-30bf-9b75-8fca9ea2d4d0_R/BasicLTI.xml I_6cc7beed-a82a-30f4-88b6-804a72a03c3a_1_R/BasicLTI.xml I_6cc7beed-a82a-30f4-88b6-804a72a03c3a_3_R/BasicLTI.xml I_6cd5ae3e-f988-3699-b5cc-985fef84f537_R/BasicLTI.xml I_6cfac659-19fd-365b-ac94-a332f6db0031_1_R/BasicLTI.xml I_6cfac659-19fd-365b-ac94-a332f6db0031_3_R/BasicLTI.xml I_6d0915df-d11c-3268-8589-6af7f6921a0c_R/BasicLTI.xml I_6d1a900b-9795-32bc-be2a-dcca5d48e2ad_1_R/BasicLTI.xml I_6d200dc0-adcc-3364-9cbb-69a659214f65_1_R/BasicLTI.xml I_6d2f02db-c629-3345-85b0-184ede1a1d9a_1_R/BasicLTI.xml I_6d385a98-1938-3b83-a565-e5453031bb54_1_R/BasicLTI.xml I_6d3fc9e3-a1f9-3df9-8022-4d6a17c120f8_R/BasicLTI.xml I_6d77922e-e181-3035-9302-a60a4015c880_1_R/BasicLTI.xml I_6d94050d-a0c0-3302-96c2-231dd5d04f03_R/BasicLTI.xml I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_1_R/BasicLTI.xml I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_3_R/BasicLTI.xml I_6d961cad-b068-3bdb-8d5d-88db12ba2a6f_5_R/BasicLTI.xml I_6dbb7a91-64de-3be2-8e5f-2e715c8d7e9a_R/BasicLTI.xml I_6dbcf5eb-0048-3133-ab7e-afce3f36b416_R/BasicLTI.xml I_6dc89e1d-b9dc-35a3-830d-e5c4a4fd5c58_1_R/BasicLTI.xml I_6dd31c25-c7fa-3338-b0af-f9bc5e30f330_R/BasicLTI.xml I_6ddc84a9-4549-3f03-af05-788125d6a052_1_R/BasicLTI.xml I_6de35046-93da-3bdf-8a12-bfd59e08e92d_R/BasicLTI.xml I_6df15052-c16e-3c29-9a1e-d8c6bfd1f4b9_1_R/BasicLTI.xml I_6e090c87-33b4-38aa-b143-d0d1f9006615_1_R/BasicLTI.xml I_6e106e01-9794-35d0-aab5-19d8496abc0b_1_R/BasicLTI.xml I_6e18a068-6015-3b51-aad6-9089f553f231_1_R/BasicLTI.xml I_6e1b919b-5a43-3d55-8997-c408a160b306_1_R/BasicLTI.xml I_6e1b919b-5a43-3d55-8997-c408a160b306_3_R/BasicLTI.xml I_6e245540-1b2c-39be-bf26-b1f4a702a239_1_R/BasicLTI.xml I_6e245540-1b2c-39be-bf26-b1f4a702a239_3_R/BasicLTI.xml I_6e2791a4-6611-3dd2-8bf6-2876d9643dcb_1_R/BasicLTI.xml I_6e2791a4-6611-3dd2-8bf6-2876d9643dcb_3_R/BasicLTI.xml I_6e297fc0-1f2b-3144-a474-11183f5b3e0e_1_R/BasicLTI.xml I_6e2ea41d-4f57-31eb-a198-4bbe1dbad5c7_1_R/BasicLTI.xml I_6e44b62a-803c-363c-8d5b-ec74c6f6c4a0_1_R/BasicLTI.xml I_6e499c33-3985-3f57-9e4e-4d0851891ee0_R/BasicLTI.xml I_6e58a518-4a7a-3235-98be-1f3dd4cc7123_R/BasicLTI.xml I_6e7f3146-a149-3699-9c1b-a08e1a1ae323_1_R/BasicLTI.xml I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_1_R/BasicLTI.xml I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_3_R/BasicLTI.xml I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_5_R/BasicLTI.xml I_6e8305bd-3652-3d23-88a5-4519b9ed6bbd_7_R/BasicLTI.xml I_6e87f8fe-1356-38a7-8941-89bb5a71c64b_1_R/BasicLTI.xml I_6e87f8fe-1356-38a7-8941-89bb5a71c64b_3_R/BasicLTI.xml I_6e9d493c-28b9-36cf-bd7f-f7136b28c636_1_R/BasicLTI.xml I_6e9d493c-28b9-36cf-bd7f-f7136b28c636_3_R/BasicLTI.xml I_6ea5e5bc-d990-3009-aca5-35263d79c50a_R/BasicLTI.xml I_6eae014b-1dfc-3480-9741-fc0bee871e89_1_R/BasicLTI.xml I_6eb520e3-8dd2-3ba5-9a00-96bd2f0db30b_1_R/BasicLTI.xml I_6eb520e3-8dd2-3ba5-9a00-96bd2f0db30b_3_R/BasicLTI.xml I_6ebc1383-1eeb-317b-ac6c-db59a0820398_1_R/BasicLTI.xml I_6eefca33-d902-3069-9075-3be2d85e6fed_R/BasicLTI.xml I_6f31d15b-e45a-3351-aabf-9113fbafffbd_1_R/BasicLTI.xml I_6f3df341-68e2-3e6f-bb51-6773dd721586_1_R/BasicLTI.xml I_6f5dae7b-f5a0-31b0-a895-e727e02d8a70_1_R/BasicLTI.xml I_6f5e06d4-86e6-3fd9-8035-9603ad7b54ff_1_R/BasicLTI.xml I_6f77334e-6e3e-3b3b-81bc-c12ddccf6c8e_1_R/BasicLTI.xml I_6f7ef230-17dc-363e-ae91-884d34cabf8f_1_R/BasicLTI.xml I_6f8dbe59-afb0-3d4b-878b-76f5945b37b4_1_R/BasicLTI.xml I_6f8f3d33-576c-34f2-b767-55d657949542_1_R/BasicLTI.xml I_6f8f3d33-576c-34f2-b767-55d657949542_3_R/BasicLTI.xml I_6f8f3d33-576c-34f2-b767-55d657949542_5_R/BasicLTI.xml I_6fd097e1-ee9a-3c55-94fe-10106b09d7b2_R/BasicLTI.xml I_6fd5cdaf-19dc-31f8-8108-481a5c203739_1_R/BasicLTI.xml I_702aadb4-fd5a-3309-9a15-377289e8e783_1_R/BasicLTI.xml I_702aadb4-fd5a-3309-9a15-377289e8e783_3_R/BasicLTI.xml I_7041f50a-817d-3cb7-9ff5-ca7d7af95ad1_R/BasicLTI.xml I_7048f0db-3f69-39e3-92de-0710383eb22d_1_R/BasicLTI.xml I_7072ce60-a9ea-33b8-b65e-7dae0360aff2_R/BasicLTI.xml I_7078f00a-0378-33d6-9d54-02688fbdba04_1_R/BasicLTI.xml I_709655a8-f4b8-33df-93e7-24b1fc9b9ad9_1_R/BasicLTI.xml I_709ad331-cdf7-3e56-b237-741e176179e4_R/BasicLTI.xml I_70a43eac-4b07-3833-a515-b4ad252246de_1_R/BasicLTI.xml I_70a63342-6a84-3e78-abd4-c2f7f2c7ebb6_1_R/BasicLTI.xml I_70a63342-6a84-3e78-abd4-c2f7f2c7ebb6_3_R/BasicLTI.xml I_70a9639d-e3bf-3172-b5ae-9bebd2c95af6_1_R/BasicLTI.xml I_70c4b034-c64d-380e-82be-d60b05147919_1_R/BasicLTI.xml I_70ee6028-874c-38c5-8047-5a6119a38c72_1_R/BasicLTI.xml I_70f194a7-1b21-3b06-bb28-856c24500feb_R/BasicLTI.xml I_71041e7c-b233-3b41-bb35-6761049a980b_R/BasicLTI.xml I_711013f3-561c-324b-a8b9-669478719708_R/BasicLTI.xml I_71172c0e-0166-3ab0-90d0-7ae741237609_1_R/BasicLTI.xml I_7117b8a5-fd86-35d1-8289-c4dc7ae4ea45_1_R/BasicLTI.xml I_7117b8a5-fd86-35d1-8289-c4dc7ae4ea45_3_R/BasicLTI.xml I_711893f8-464f-3ab7-bfd7-cb915b4af72d_1_R/BasicLTI.xml I_712b0357-7bee-3595-b59e-c37f48bdb877_R/BasicLTI.xml I_7147c5d7-4764-32b7-a8a0-bc8f21ca852e_R/BasicLTI.xml I_7148bf95-638c-3f5d-b096-03c9698cf90d_1_R/BasicLTI.xml I_714fe672-5301-3f4d-aec9-82e4eaaf62ef_R/BasicLTI.xml I_7169d829-d2f6-3187-96f1-30688793f4cf_R/BasicLTI.xml I_716d8cf1-fb64-3bbb-9778-4c26e7dedd38_1_R/BasicLTI.xml I_716d8cf1-fb64-3bbb-9778-4c26e7dedd38_3_R/BasicLTI.xml I_716ef1f8-de04-301a-b586-14fa7efd984a_R/BasicLTI.xml I_718adabc-8f8b-3f35-9ca7-75995c9eda0f_1_R/BasicLTI.xml I_718adabc-8f8b-3f35-9ca7-75995c9eda0f_3_R/BasicLTI.xml I_7193a844-9608-3216-a31f-bf508b4fe3c1_R/BasicLTI.xml I_7195c306-5db4-336f-a715-a30d4b8b34e9_R/BasicLTI.xml I_71a089e6-5480-3002-a977-05ea387a559a_1_R/BasicLTI.xml I_71cb2eab-df88-3c2f-be80-56205d185827_1_R/BasicLTI.xml I_71cd562d-b028-37f4-9948-b259b0fc0cbe_1_R/BasicLTI.xml I_71da1241-816d-3781-9a83-f8ac42a2bb93_1_R/BasicLTI.xml I_71da1241-816d-3781-9a83-f8ac42a2bb93_3_R/BasicLTI.xml I_71da1241-816d-3781-9a83-f8ac42a2bb93_5_R/BasicLTI.xml I_71ec2c94-8ebf-3974-a20b-efd7fcffe3af_R/BasicLTI.xml I_71f00174-364d-306c-842b-bb36284d4338_1_R/BasicLTI.xml I_72135da7-d410-3aba-a825-64b352199f98_R/BasicLTI.xml I_721e4198-db44-30f3-8214-acface76233f_1_R/BasicLTI.xml I_722b7a74-4869-3b50-a529-70eec1aceae6_1_R/BasicLTI.xml I_722d717e-c424-30d5-bfa9-150570db5aee_1_R/BasicLTI.xml I_72629c80-2bd7-362c-b425-595b111bafbb_1_R/BasicLTI.xml I_7281a39b-bdfc-35c7-9df8-f931d754d099_1_R/BasicLTI.xml I_72b03a81-1900-3814-aea7-622861b6a325_1_R/BasicLTI.xml I_72b4461c-4d77-3bfb-a459-95007323167f_R/BasicLTI.xml I_72c44d7f-39de-3429-8249-b545a3815d45_1_R/BasicLTI.xml I_72c44d7f-39de-3429-8249-b545a3815d45_3_R/BasicLTI.xml I_72d861c8-10fb-3766-bdaf-ae6029c90ad3_R/BasicLTI.xml I_72e13228-42b4-3c31-becb-7b83ee9ad72e_1_R/BasicLTI.xml I_72e774f7-9ca8-302e-9f14-a8caacc4fd3b_R/BasicLTI.xml I_731afb9d-fb83-3258-9874-d23aa9405060_R/BasicLTI.xml I_733ad4dd-dacc-3f73-a276-8e1064792036_R/BasicLTI.xml I_734ca048-9e58-380e-96cb-01b0dff763a9_1_R/BasicLTI.xml I_736d9058-a605-3df6-b614-fa605cb2dbb1_1_R/BasicLTI.xml I_737603fb-92b8-33e2-9464-96a97bbc3230_1_R/BasicLTI.xml I_738b5cc4-bee3-31ab-8af0-c37f4d699942_1_R/BasicLTI.xml I_7397bbae-7b92-3c71-9ea0-c17ff6c11346_1_R/BasicLTI.xml I_739e1ca9-bb6a-3c25-a3ab-aaedeb726694_1_R/BasicLTI.xml I_739e1ca9-bb6a-3c25-a3ab-aaedeb726694_3_R/BasicLTI.xml I_73b236a4-498b-3e03-8294-f77dbaf25a57_1_R/BasicLTI.xml I_73df9dbb-510f-37b9-b5d4-f2f1ea3e89c4_1_R/BasicLTI.xml I_73e0cb82-c093-3597-85d6-e3539938cd74_1_R/BasicLTI.xml I_73e0cb82-c093-3597-85d6-e3539938cd74_3_R/BasicLTI.xml I_73e0cb82-c093-3597-85d6-e3539938cd74_5_R/BasicLTI.xml I_73e0cb82-c093-3597-85d6-e3539938cd74_7_R/BasicLTI.xml I_73ea0616-8f9d-38d1-a206-0afc3d752e5e_1_R/BasicLTI.xml I_73eb323e-8c4e-3cf2-b26d-e95245a21736_1_R/BasicLTI.xml I_73eb323e-8c4e-3cf2-b26d-e95245a21736_3_R/BasicLTI.xml I_73eb323e-8c4e-3cf2-b26d-e95245a21736_5_R/BasicLTI.xml I_73eb323e-8c4e-3cf2-b26d-e95245a21736_7_R/BasicLTI.xml I_7404c1b5-0f91-3871-adbc-ffadc7a80165_R/BasicLTI.xml I_7408e98c-0eca-3ade-9459-9f2ed883dd57_1_R/BasicLTI.xml I_7428b87f-9b07-3f56-a9e6-fd38634c011e_1_R/BasicLTI.xml I_7428b87f-9b07-3f56-a9e6-fd38634c011e_3_R/BasicLTI.xml I_743eb699-d26e-3222-92be-fd3f723ab740_R/BasicLTI.xml I_743f12c8-fec4-35e2-aff8-c5eb4124d945_1_R/BasicLTI.xml I_743f12c8-fec4-35e2-aff8-c5eb4124d945_3_R/BasicLTI.xml I_743f12c8-fec4-35e2-aff8-c5eb4124d945_5_R/BasicLTI.xml I_74456bea-4850-3d1c-b278-2de8114c4a80_1_R/BasicLTI.xml I_74456bea-4850-3d1c-b278-2de8114c4a80_3_R/BasicLTI.xml I_74456bea-4850-3d1c-b278-2de8114c4a80_5_R/BasicLTI.xml I_7458f054-ed11-38e6-8840-a63536bcf209_1_R/BasicLTI.xml I_7458f054-ed11-38e6-8840-a63536bcf209_3_R/BasicLTI.xml I_74720486-ab6c-39e1-b26e-ca80c3013d47_1_R/BasicLTI.xml I_749cf6df-29d5-3aaf-a296-131aa24866ef_1_R/BasicLTI.xml I_74cc4343-2de8-3e64-bdb5-04a2d39ac16f_1_R/BasicLTI.xml I_74ced5b1-cbb4-3f68-9555-8fc569fd6dbf_1_R/BasicLTI.xml I_74f8113b-ccca-384e-b027-0469b1309ca1_1_R/BasicLTI.xml I_74fed7f3-a25c-3c97-a51f-7dcb0e6d24c8_1_R/BasicLTI.xml I_74fed7f3-a25c-3c97-a51f-7dcb0e6d24c8_3_R/BasicLTI.xml I_74fed7f3-a25c-3c97-a51f-7dcb0e6d24c8_5_R/BasicLTI.xml I_74fed7f3-a25c-3c97-a51f-7dcb0e6d24c8_7_R/BasicLTI.xml I_74fed7f3-a25c-3c97-a51f-7dcb0e6d24c8_9_R/BasicLTI.xml I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_1_R/BasicLTI.xml I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_3_R/BasicLTI.xml I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_5_R/BasicLTI.xml I_751953ed-cbcb-3bf6-80a4-b0102d765c4d_7_R/BasicLTI.xml I_75229e48-78bf-384a-826e-4c0cc6f53a12_1_R/BasicLTI.xml I_75298b72-8419-3b8a-8d93-c9955ea03fc3_1_R/BasicLTI.xml I_75324ab7-fad1-3199-88c1-7dfdf4555791_1_R/BasicLTI.xml I_75384fa3-d57a-343e-90e6-a1fa39660889_1_R/BasicLTI.xml I_753ce06f-a585-3e89-9ff4-8f32e846d8a6_1_R/BasicLTI.xml I_75504ad3-2474-37e5-8d18-d633aa1d18a3_1_R/BasicLTI.xml I_7568df37-24f3-3341-94c2-e8b8a32c4661_1_R/BasicLTI.xml I_7569b35f-ec21-3b2e-93df-07042efe27e2_R/BasicLTI.xml I_757296b6-4045-32a8-854d-ef551cca2f4e_R/BasicLTI.xml I_7572c324-73ca-375a-a927-d15a05bf97a8_1_R/BasicLTI.xml I_759c0da5-302b-3036-8891-c7cec954b88e_R/BasicLTI.xml I_75d396c5-04c0-3ca7-8224-60f5341c76c0_1_R/BasicLTI.xml I_75d396c5-04c0-3ca7-8224-60f5341c76c0_3_R/BasicLTI.xml I_75d7d284-02eb-3399-ab7f-4dd57d65c203_R/BasicLTI.xml I_75ed7e4c-1e7e-3d17-b25b-f080940db0ed_1_R/BasicLTI.xml I_76029f19-41e2-32d0-bfa0-f4f84a5d0859_1_R/BasicLTI.xml I_76029f19-41e2-32d0-bfa0-f4f84a5d0859_3_R/BasicLTI.xml I_7610e6bd-8b91-34ef-92f3-724eb3d82cbe_R/BasicLTI.xml I_7610f352-36e1-3561-bd61-8ca688883104_R/BasicLTI.xml I_763fbba5-4d99-3203-9bce-4c545adfb626_1_R/BasicLTI.xml I_7659a2dd-2583-3bcf-9e6f-b738449d9958_R/BasicLTI.xml I_76698b2a-d7a3-3d52-9094-f9adc392a93d_1_R/BasicLTI.xml I_7674ce66-1a02-3d51-a1be-591fed1a44bb_R/BasicLTI.xml I_768ce2f2-f463-31e8-979a-b0f773097864_1_R/BasicLTI.xml I_76b82bf6-e0b2-3733-a8b4-65175e9b0cb1_1_R/BasicLTI.xml I_76be449a-761c-376a-a2a0-3d00495a8a51_1_R/BasicLTI.xml I_76be449a-761c-376a-a2a0-3d00495a8a51_3_R/BasicLTI.xml I_76be449a-761c-376a-a2a0-3d00495a8a51_5_R/BasicLTI.xml I_76be449a-761c-376a-a2a0-3d00495a8a51_7_R/BasicLTI.xml I_76c1a411-6dee-3c3e-9c2b-2d2aff1762af_R/BasicLTI.xml I_76c24827-b781-32df-9997-af9aeac2cbd7_R/BasicLTI.xml I_76e17fbf-56c7-3f47-b42d-fae724941c07_R/BasicLTI.xml I_76e450ee-bcdb-3fad-b1b5-01c6e9990690_1_R/BasicLTI.xml I_76e8ee7c-f812-34cb-bf4f-18d2fc34f7fe_1_R/BasicLTI.xml I_77054909-f87f-35df-9ee2-4154925b99bd_1_R/BasicLTI.xml I_770bfcfe-d4a6-33bd-afac-67c8c54e754c_1_R/BasicLTI.xml I_770bfcfe-d4a6-33bd-afac-67c8c54e754c_3_R/BasicLTI.xml I_770cc26c-a1f4-3871-b282-88b258b6258c_R/BasicLTI.xml I_7737b1cf-080e-3a7c-b9fd-e06f973489a5_R/BasicLTI.xml I_7748b531-0a25-3c5e-8f47-c8d3003182ba_1_R/BasicLTI.xml I_77499fd7-62ab-3ab8-a1d5-7577624c88a1_R/BasicLTI.xml I_775456d9-d5d8-351b-a211-bbc8ad0d65a3_R/BasicLTI.xml I_7767e499-cd03-3480-923f-0cfe46e0e9e4_1_R/BasicLTI.xml I_779a9a31-746b-3f66-9a5e-fd96af25d9c0_1_R/BasicLTI.xml I_77a4a8d9-2d79-3ee0-89b9-e1162f6fb2a4_R/BasicLTI.xml I_77da2239-318f-304f-9c32-615267d83c40_1_R/BasicLTI.xml I_7800dd33-a29f-380f-b21e-44d0460e9f6b_1_R/BasicLTI.xml I_7800dd33-a29f-380f-b21e-44d0460e9f6b_3_R/BasicLTI.xml I_7800dd33-a29f-380f-b21e-44d0460e9f6b_5_R/BasicLTI.xml I_7800dd33-a29f-380f-b21e-44d0460e9f6b_7_R/BasicLTI.xml I_7823623e-64e1-33aa-9da0-2fd301882089_1_R/BasicLTI.xml I_7823623e-64e1-33aa-9da0-2fd301882089_3_R/BasicLTI.xml I_7823623e-64e1-33aa-9da0-2fd301882089_5_R/BasicLTI.xml I_7823623e-64e1-33aa-9da0-2fd301882089_7_R/BasicLTI.xml I_7823623e-64e1-33aa-9da0-2fd301882089_9_R/BasicLTI.xml I_78408c7e-fb22-3fdc-9364-69be951c033a_1_R/BasicLTI.xml I_78417c24-2bd9-333c-8d82-04e42c605492_1_R/BasicLTI.xml I_785ce2ac-8d5e-31eb-a241-95732da9e5e3_R/BasicLTI.xml I_78613273-ea6f-33aa-a3ba-ecc2b6ecdc1c_1_R/BasicLTI.xml I_7888047c-9b86-3d15-804c-750c85468309_1_R/BasicLTI.xml I_7894811a-4701-3ac0-9c0b-a3b6bcdf3877_1_R/BasicLTI.xml I_7896de19-af7e-3fe5-81b3-039ea0cf7f16_R/BasicLTI.xml I_789a9b98-9e5d-379a-96a8-d457cb48657f_1_R/BasicLTI.xml I_78a26116-f4e0-3d07-9eba-9f8f0e17c433_R/BasicLTI.xml I_78a79a18-e07b-39ea-a04b-0435b16235f1_R/BasicLTI.xml I_78a97d02-af32-35d7-a755-fce1b57dcbd8_1_R/BasicLTI.xml I_78d2c8a3-5363-3578-8bb2-5c18ca2c308a_1_R/BasicLTI.xml I_78d77537-012d-33db-b7de-a417e882a88c_1_R/BasicLTI.xml I_78d77537-012d-33db-b7de-a417e882a88c_3_R/BasicLTI.xml I_78d77537-012d-33db-b7de-a417e882a88c_5_R/BasicLTI.xml I_78d77537-012d-33db-b7de-a417e882a88c_7_R/BasicLTI.xml I_78d77537-012d-33db-b7de-a417e882a88c_9_R/BasicLTI.xml I_78e0ed4f-6b21-31bc-a4a8-daab30ec9c8e_1_R/BasicLTI.xml I_78e0ed4f-6b21-31bc-a4a8-daab30ec9c8e_3_R/BasicLTI.xml I_78e0ed4f-6b21-31bc-a4a8-daab30ec9c8e_5_R/BasicLTI.xml I_78e0ed4f-6b21-31bc-a4a8-daab30ec9c8e_7_R/BasicLTI.xml I_78e0ed4f-6b21-31bc-a4a8-daab30ec9c8e_9_R/BasicLTI.xml I_78e22690-101e-3541-a7e8-9fe7c398face_1_R/BasicLTI.xml I_78f62449-7c69-35bc-b1cc-66b8e86068fb_1_R/BasicLTI.xml I_791a89aa-38d3-343c-8600-cc3110f3445e_1_R/BasicLTI.xml I_791a89aa-38d3-343c-8600-cc3110f3445e_3_R/BasicLTI.xml I_791a89aa-38d3-343c-8600-cc3110f3445e_5_R/BasicLTI.xml I_791a89aa-38d3-343c-8600-cc3110f3445e_7_R/BasicLTI.xml I_791a89aa-38d3-343c-8600-cc3110f3445e_9_R/BasicLTI.xml I_791d7d7e-484d-314c-826e-15d34b6796bc_1_R/BasicLTI.xml I_791e54c0-c958-37a9-8807-b1b76c209e05_R/BasicLTI.xml I_793c1b72-c331-3140-a4b3-379445e8322e_1_R/BasicLTI.xml I_793dbb52-066e-3112-8c33-96dfbd593745_1_R/BasicLTI.xml I_793dbb52-066e-3112-8c33-96dfbd593745_3_R/BasicLTI.xml I_793dbb52-066e-3112-8c33-96dfbd593745_5_R/BasicLTI.xml I_793dbb52-066e-3112-8c33-96dfbd593745_7_R/BasicLTI.xml I_793dbb52-066e-3112-8c33-96dfbd593745_9_R/BasicLTI.xml I_798562cd-dcf6-3b79-9fd9-704d640398d2_R/BasicLTI.xml I_79c38c62-963c-35b2-8e82-10dcdc7456b6_1_R/BasicLTI.xml I_79e15c4d-d4af-3c1a-ba03-7ff3fd6556e1_R/BasicLTI.xml I_79e30f39-f6f7-37d1-b34f-c2a0e43dcfe9_1_R/BasicLTI.xml I_7a1ac939-5564-3b8d-9e7e-b0092529a1df_R/BasicLTI.xml I_7a1eecfb-abbb-3313-8801-10eef72ae8c1_1_R/BasicLTI.xml I_7a2a6581-ebe0-367f-91e5-9fd174bacce2_1_R/BasicLTI.xml I_7a2a6581-ebe0-367f-91e5-9fd174bacce2_3_R/BasicLTI.xml I_7a30da3f-a76d-363c-a79b-6aeb1c8dfb59_1_R/BasicLTI.xml I_7a3a23d5-4c0c-36e1-ab39-db1689a0de0b_R/BasicLTI.xml I_7a3e37a0-1051-3591-ad0d-9178f5ad88ec_1_R/BasicLTI.xml I_7a5eddfa-9574-33ed-84fa-d3a8e64048fb_1_R/BasicLTI.xml I_7a5eddfa-9574-33ed-84fa-d3a8e64048fb_3_R/BasicLTI.xml I_7a5eddfa-9574-33ed-84fa-d3a8e64048fb_5_R/BasicLTI.xml I_7a682f05-19ed-3b17-91ef-fd9ed73f6d19_1_R/BasicLTI.xml I_7a761553-f517-3bb0-b585-a74508f6404a_1_R/BasicLTI.xml I_7a761553-f517-3bb0-b585-a74508f6404a_3_R/BasicLTI.xml I_7a796807-5207-3ef3-959c-cb2c71365103_1_R/BasicLTI.xml I_7a949ec8-91bc-3cbc-a91c-83f252fb564a_1_R/BasicLTI.xml I_7a98b69b-75ad-38a9-87c0-856b08f062ec_R/BasicLTI.xml I_7ac78de8-cf0b-3284-9d80-c2acf383c01b_1_R/BasicLTI.xml I_7ad90de5-f33b-3555-8afc-3c35d084975a_1_R/BasicLTI.xml I_7ad90de5-f33b-3555-8afc-3c35d084975a_3_R/BasicLTI.xml I_7ad90de5-f33b-3555-8afc-3c35d084975a_5_R/BasicLTI.xml I_7ad90de5-f33b-3555-8afc-3c35d084975a_7_R/BasicLTI.xml I_7adbc991-b0b6-3d01-92a6-b695480190eb_1_R/BasicLTI.xml I_7b0ad373-968c-3376-8fa8-72d08e05b065_1_R/BasicLTI.xml I_7b1ad2ab-f4ee-3bd2-8f07-3b9bce20415d_R/BasicLTI.xml I_7b1ee5dc-337f-307a-a660-d40a37099759_1_R/BasicLTI.xml I_7b5d2312-142d-3a4e-b6f9-2b3818903de7_R/BasicLTI.xml I_7b6aac91-a3c7-3e2b-9cb0-e7fa53e64235_R/BasicLTI.xml I_7b83351f-63d4-3c71-bb82-e5adc75232cc_1_R/BasicLTI.xml I_7b83c5b5-cab4-3b8c-b60e-7a53ef125808_R/BasicLTI.xml I_7b845a71-43dd-31da-9dcb-5ccd6dbf69c7_R/BasicLTI.xml I_7b989124-b6a3-3a9a-bc46-5c387a785058_R/BasicLTI.xml I_7bdc09dd-2c57-32a4-a286-c1f85eaa8ef9_1_R/BasicLTI.xml I_7c04d2af-7e40-3d3a-b18c-2b7e1d6bc2d1_R/BasicLTI.xml I_7c100b3d-2bae-3b7a-a56f-12176f0c2440_1_R/BasicLTI.xml I_7c173e0d-b22c-398f-8342-a0f26db1b8ef_1_R/BasicLTI.xml I_7c18bb6d-e7b0-3aa3-8d39-3073cf6f27c2_R/BasicLTI.xml I_7c2019c4-9535-3df0-9a34-0499beb8bdb1_1_R/BasicLTI.xml I_7c3597a3-9fa4-3c1e-9c60-5bf51fe0dfb0_R/BasicLTI.xml I_7c3a5744-e13f-3412-a04a-1a0e270b4538_1_R/BasicLTI.xml I_7c4fd4df-2252-351f-9b8c-6752a43da278_R/BasicLTI.xml I_7c576142-e277-3cba-83af-9874c3cb938d_1_R/BasicLTI.xml I_7c6a15e9-935c-3ccb-928a-56dd18198c71_R/BasicLTI.xml I_7c7411da-65ef-30a3-af95-13400828e40d_1_R/BasicLTI.xml I_7c977dd8-811d-3c9b-ad05-ebe8dffafaf2_1_R/BasicLTI.xml I_7c98a7f6-8ce4-366e-8eab-7e09fdf04d0e_1_R/BasicLTI.xml I_7c9951b8-5784-3793-a289-1303a3be251d_1_R/BasicLTI.xml I_7c9951b8-5784-3793-a289-1303a3be251d_3_R/BasicLTI.xml I_7cb21154-5112-33c3-8265-88f0276f22d7_1_R/BasicLTI.xml I_7ccda6a5-519c-3624-a2ea-15fbfb4f42c2_1_R/BasicLTI.xml I_7cdbb1de-d874-312e-a121-500cb4fe35af_1_R/BasicLTI.xml I_7cf87c42-86f4-3003-8918-542a98eb29a4_R/BasicLTI.xml I_7cfadc04-d0bd-32da-9c89-8fd0ebe62002_1_R/BasicLTI.xml I_7cfc1b63-030d-37ed-8789-4ca4a79c202f_R/BasicLTI.xml I_7cfdbe93-c35e-3d58-8dbc-f9653be60a34_R/BasicLTI.xml I_7d0a4032-7454-3124-832a-366e138247e5_R/BasicLTI.xml I_7d16e74a-753c-36ce-89b7-033a935a8489_1_R/BasicLTI.xml I_7d176921-6ccc-310b-a2a8-c815eae5535d_R/BasicLTI.xml I_7d21f6ff-ddb1-3e15-8588-7a6ab3f133fb_1_R/BasicLTI.xml I_7d21f6ff-ddb1-3e15-8588-7a6ab3f133fb_3_R/BasicLTI.xml I_7d268985-e7c1-3a02-b557-5f478a0200f4_1_R/BasicLTI.xml I_7d401470-d968-3fd7-a1a9-06c3a4343bfb_R/BasicLTI.xml I_7d4a3a5a-8567-309e-9069-9a15daa23c01_1_R/BasicLTI.xml I_7d5176d7-6da3-313d-8769-d3aecf1fe786_R/BasicLTI.xml I_7d654def-c378-3233-8d11-a818dffec879_1_R/BasicLTI.xml I_7d6c25fa-08e3-3806-8b1d-aa34785cb576_1_R/BasicLTI.xml I_7d6c25fa-08e3-3806-8b1d-aa34785cb576_3_R/BasicLTI.xml I_7d6c25fa-08e3-3806-8b1d-aa34785cb576_5_R/BasicLTI.xml I_7d71a338-fbcd-330e-b328-1de48eb64392_R/BasicLTI.xml I_7d93cb5a-6f35-3887-bc3f-25af29610a1c_R/BasicLTI.xml I_7da27b34-914f-3978-a86c-62aabe54daef_1_R/BasicLTI.xml I_7dc5c610-b309-3cd0-bc86-943b9ebcba99_R/BasicLTI.xml I_7de000dc-19db-3fc1-90f4-bee2d11f23d7_R/BasicLTI.xml I_7de1dcad-03b0-30ff-84e8-d897cf3c4e71_1_R/BasicLTI.xml I_7de1dcad-03b0-30ff-84e8-d897cf3c4e71_3_R/BasicLTI.xml I_7de1dcad-03b0-30ff-84e8-d897cf3c4e71_5_R/BasicLTI.xml I_7de1dcad-03b0-30ff-84e8-d897cf3c4e71_7_R/BasicLTI.xml I_7de1dcad-03b0-30ff-84e8-d897cf3c4e71_9_R/BasicLTI.xml I_7de83599-a5a9-3a6f-9ac5-06fbf46a7247_1_R/BasicLTI.xml I_7dfb7bf6-ea96-35f3-8570-e8611f21f90d_1_R/BasicLTI.xml I_7dfe5732-5fcb-3439-857f-9e61f7a4907a_1_R/BasicLTI.xml I_7dfe5732-5fcb-3439-857f-9e61f7a4907a_3_R/BasicLTI.xml I_7e00f08c-0c77-3dc7-a772-f0f0cb581fca_1_R/BasicLTI.xml I_7e05b22b-2da2-31d7-ad39-55905e4acf20_R/BasicLTI.xml I_7e09dcc6-abb9-3bbe-ac24-d4ea8111823e_R/BasicLTI.xml I_7e0c2839-f5fa-340b-888b-b7bb697c20cd_R/BasicLTI.xml I_7e0e68d1-f708-34e6-9cfc-dd037d3fbba4_1_R/BasicLTI.xml I_7e132fdc-8f00-30e4-a44b-c90a20167689_1_R/BasicLTI.xml I_7e27e4b4-b253-3c66-8339-9fd692472ea3_1_R/BasicLTI.xml I_7e27e4b4-b253-3c66-8339-9fd692472ea3_3_R/BasicLTI.xml I_7e27e4b4-b253-3c66-8339-9fd692472ea3_5_R/BasicLTI.xml I_7e27e4b4-b253-3c66-8339-9fd692472ea3_7_R/BasicLTI.xml I_7e30f261-e8ef-353b-b5b2-f827b17761ba_1_R/BasicLTI.xml I_7e30f261-e8ef-353b-b5b2-f827b17761ba_3_R/BasicLTI.xml I_7e3a7f3b-481e-3864-a58d-2f0ce30c234b_1_R/BasicLTI.xml I_7e4363b1-afa2-3e8b-8669-cb951ef7c1c5_1_R/BasicLTI.xml I_7e4363b1-afa2-3e8b-8669-cb951ef7c1c5_3_R/BasicLTI.xml I_7e64e63d-a864-3669-9955-98ece241c1ca_1_R/BasicLTI.xml I_7e8940e3-7c92-371d-a56b-854da3a52add_1_R/BasicLTI.xml I_7e8940e3-7c92-371d-a56b-854da3a52add_3_R/BasicLTI.xml I_7e8940e3-7c92-371d-a56b-854da3a52add_5_R/BasicLTI.xml I_7e8940e3-7c92-371d-a56b-854da3a52add_7_R/BasicLTI.xml I_7e8b2096-1de5-3f15-a5ec-4b642eaf68c1_R/BasicLTI.xml I_7e92a89e-9b86-34bf-9f47-690dbfcc6403_1_R/BasicLTI.xml I_7e9587b4-eb37-3aec-b1cf-4d58e144a132_R/BasicLTI.xml I_7e96122d-aa09-3e16-ac7e-d14678ce8d73_R/BasicLTI.xml I_7eab86fb-d9ce-3313-ae38-29b78924cb3d_R/BasicLTI.xml I_7eca3686-91fe-33c8-a58e-0b01a558854d_R/BasicLTI.xml I_7eff3db4-794d-307b-8b47-0b0af5a23d31_1_R/BasicLTI.xml I_7f0941bc-afc9-349b-96a3-2a6f1fd50ad1_1_R/BasicLTI.xml I_7f1687e1-9b3c-3355-8a67-4f335b5b2879_R/BasicLTI.xml I_7f27e8e9-11dd-392e-b130-3bb06bc089ec_R/BasicLTI.xml I_7f2a3373-d969-359e-b2be-da9de945bbb5_R/BasicLTI.xml I_7f37e1a9-d201-39d6-beb6-0dabce4a9506_1_R/BasicLTI.xml I_7f412372-f747-3531-8610-6c7203677a4b_1_R/BasicLTI.xml I_7f5e920f-e81a-3419-8d7c-528b4b3d1549_1_R/BasicLTI.xml I_7f6996e1-b00f-3847-9db8-c867c5abd888_1_R/BasicLTI.xml I_7f6a9c3b-55e0-3c40-bb88-2745b8dd8684_1_R/BasicLTI.xml I_7fa222b1-e5d0-3d05-9a74-a41f05b61719_R/BasicLTI.xml I_7fdd7812-4679-3b90-9e84-93e68e749b23_R/BasicLTI.xml I_7ff61cbd-549b-3bf9-87f9-84171a27e638_R/BasicLTI.xml I_7ff99b32-6429-3b41-9389-93939d61cebc_1_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_11_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_1_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_3_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_5_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_7_R/BasicLTI.xml I_80029533-0e8c-3888-a516-697ca1e46948_9_R/BasicLTI.xml I_80262396-7327-3b71-8349-1c45d6d2d76a_1_R/BasicLTI.xml I_80262396-7327-3b71-8349-1c45d6d2d76a_3_R/BasicLTI.xml I_80284070-81dc-3535-9bb3-82f854e975b1_1_R/BasicLTI.xml I_80284070-81dc-3535-9bb3-82f854e975b1_3_R/BasicLTI.xml I_802db03a-507a-33d0-bf80-cf5ca8d49fcb_R/BasicLTI.xml I_803063c1-7105-3f0f-baad-e9380cbd18f1_1_R/BasicLTI.xml I_803063c1-7105-3f0f-baad-e9380cbd18f1_3_R/BasicLTI.xml I_803ae06c-aec9-30d7-8fdf-e4d747d1a9c2_1_R/BasicLTI.xml I_8050b2de-f03c-3d81-a603-f7e5654c1956_1_R/BasicLTI.xml I_80574104-868a-3929-878a-7bae94aa9b56_R/BasicLTI.xml I_80598533-f48a-35f9-b9b6-9b864d63593d_1_R/BasicLTI.xml I_8093d9a8-7c31-3aee-a8ab-dceb856858f9_1_R/BasicLTI.xml I_80a4f50c-da31-3d97-a554-f6e98a19e6c0_1_R/BasicLTI.xml I_80b4ee29-a2ff-3ac1-8750-bad1c9e3066d_1_R/BasicLTI.xml I_80b4ee29-a2ff-3ac1-8750-bad1c9e3066d_3_R/BasicLTI.xml I_80b4ee29-a2ff-3ac1-8750-bad1c9e3066d_5_R/BasicLTI.xml I_80b4ee29-a2ff-3ac1-8750-bad1c9e3066d_7_R/BasicLTI.xml I_80bae5ca-3982-3bce-9f3a-be31553ebb75_R/BasicLTI.xml I_80cea7aa-fb32-338c-8133-e41a02af9200_1_R/BasicLTI.xml I_80ec1d9a-d439-3aeb-9442-a5dd50b95d3d_1_R/BasicLTI.xml I_80ec1d9a-d439-3aeb-9442-a5dd50b95d3d_3_R/BasicLTI.xml I_80ec1d9a-d439-3aeb-9442-a5dd50b95d3d_5_R/BasicLTI.xml I_80ec1d9a-d439-3aeb-9442-a5dd50b95d3d_7_R/BasicLTI.xml I_80ec1d9a-d439-3aeb-9442-a5dd50b95d3d_9_R/BasicLTI.xml I_80ffb8b0-6e91-3bb4-bd52-d1934b60733d_1_R/BasicLTI.xml I_80ffb8b0-6e91-3bb4-bd52-d1934b60733d_3_R/BasicLTI.xml I_80ffb8b0-6e91-3bb4-bd52-d1934b60733d_5_R/BasicLTI.xml I_80ffb8b0-6e91-3bb4-bd52-d1934b60733d_7_R/BasicLTI.xml I_80ffb8b0-6e91-3bb4-bd52-d1934b60733d_9_R/BasicLTI.xml I_816fa8ca-24a8-3900-b0fc-7a4eddba37ef_1_R/BasicLTI.xml I_81727ab8-6822-3250-b116-0562917fe33f_R/BasicLTI.xml I_81ad7a0e-5b08-3cf0-abf3-cfd2cc56660c_1_R/BasicLTI.xml I_81c54fba-a24c-3058-b873-3bd687e55048_R/BasicLTI.xml I_81e03ff9-bc61-3031-93d7-c81904478ef7_1_R/BasicLTI.xml I_81e6246c-2475-38ac-8d18-8211cc33d0f9_R/BasicLTI.xml I_8201e114-70c4-3e46-b9bb-94393498e524_1_R/BasicLTI.xml I_820dfba1-2b9f-3058-b4b6-fe20a9bc37e8_1_R/BasicLTI.xml I_821da87b-6623-3410-b302-a86426e6982c_1_R/BasicLTI.xml I_824cf27f-30eb-300c-bc84-8a92009f34d4_1_R/BasicLTI.xml I_824e2405-2b73-3c98-85a0-c7ff419cb6d0_1_R/BasicLTI.xml I_825593d4-7540-3c24-a2ca-2302edd123b8_1_R/BasicLTI.xml I_825b7295-1cb9-3939-ad07-1a58c3dea9fc_1_R/BasicLTI.xml I_825fb325-73bb-36c4-ba5c-2e61d555f5a1_1_R/BasicLTI.xml I_8261c91a-068a-31ee-951a-1245a79e58bd_R/BasicLTI.xml I_82738b32-a620-30f1-8c53-140c12534346_R/BasicLTI.xml I_82766433-42b9-3baa-9433-27313c3e4ae8_R/BasicLTI.xml I_82a14373-39e2-3144-83fb-567258571ddc_1_R/BasicLTI.xml I_82a14373-39e2-3144-83fb-567258571ddc_3_R/BasicLTI.xml I_82a14373-39e2-3144-83fb-567258571ddc_5_R/BasicLTI.xml I_82a14373-39e2-3144-83fb-567258571ddc_7_R/BasicLTI.xml I_82a14373-39e2-3144-83fb-567258571ddc_9_R/BasicLTI.xml I_82b461e5-8a56-39e1-a33c-27844929d893_R/BasicLTI.xml I_82ba7ee8-425f-3616-ad40-ee5ad7c58c69_1_R/BasicLTI.xml I_82c4d969-363d-3c47-a571-826d434cc072_R/BasicLTI.xml I_82cc1204-5a46-336e-93e7-28bb41af25bf_1_R/BasicLTI.xml I_82d50c47-8ddc-3cee-a707-89c64d7ed6b4_1_R/BasicLTI.xml I_82f58e39-c483-3a0a-a747-ad4f22491e5e_1_R/BasicLTI.xml I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_1_R/BasicLTI.xml I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_3_R/BasicLTI.xml I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_5_R/BasicLTI.xml I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_7_R/BasicLTI.xml I_82fbc40c-fcb7-3bec-a4d6-1ef954828ddb_9_R/BasicLTI.xml I_82ff3c3e-bfeb-3f9b-a239-7bf88bf54307_1_R/BasicLTI.xml I_83059cd4-615a-36b9-9ce1-734f16b07d87_R/BasicLTI.xml I_830b0eaf-3b86-3276-bcb9-6699e2eeb124_R/BasicLTI.xml I_832bf82d-d145-363d-a747-e76850bc69b0_R/BasicLTI.xml I_8332c60e-1f0c-39a8-a906-c3e2f66c8d23_R/BasicLTI.xml I_8335bcdd-d9f8-32ed-b393-2c781c95a530_1_R/BasicLTI.xml I_8340f9af-03e4-352f-a254-dde8713726cb_1_R/BasicLTI.xml I_834902ad-841b-3efb-a0f8-32645a17b7ce_R/BasicLTI.xml I_83516287-4824-37ab-bf97-d7d0cbc1b935_R/BasicLTI.xml I_83685acb-f909-3e08-9964-5dcdc493e075_1_R/BasicLTI.xml I_83685acb-f909-3e08-9964-5dcdc493e075_3_R/BasicLTI.xml I_83685acb-f909-3e08-9964-5dcdc493e075_5_R/BasicLTI.xml I_83711aac-c044-311e-bad1-bf347490fa5e_R/BasicLTI.xml I_83851b32-0232-3bf4-8aaa-73495e47bc03_1_R/BasicLTI.xml I_83a7d5d9-6d06-343e-be96-967cc58d405f_1_R/BasicLTI.xml I_83a7d5d9-6d06-343e-be96-967cc58d405f_3_R/BasicLTI.xml I_83b04c4d-d5de-3b6e-9cce-285d110bbd71_R/BasicLTI.xml I_83ba1195-80a7-3197-8a70-0cd6e982ca1c_1_R/BasicLTI.xml I_83c7fcd7-2d6c-3f7f-bd6f-d2cecbe4748d_1_R/BasicLTI.xml I_83df2bb6-1d85-30e9-840a-4db833813db3_1_R/BasicLTI.xml I_84158122-d84b-376e-bde9-7f0611146c41_R/BasicLTI.xml I_842edbf4-6751-309b-94f8-6e2b711cca0a_1_R/BasicLTI.xml I_8431c932-9a3a-3018-a34e-18c3b209aa2f_1_R/BasicLTI.xml I_8431c932-9a3a-3018-a34e-18c3b209aa2f_3_R/BasicLTI.xml I_8431c932-9a3a-3018-a34e-18c3b209aa2f_5_R/BasicLTI.xml I_8431c932-9a3a-3018-a34e-18c3b209aa2f_7_R/BasicLTI.xml I_84437e05-0f74-39f2-844e-7fb4df88edaf_R/BasicLTI.xml I_844f92ce-ec56-3cce-847f-6985cb867a83_1_R/BasicLTI.xml I_846137c8-0032-3589-a711-704929055a65_R/BasicLTI.xml I_84775469-b0c1-3398-845e-1b2c152956a6_1_R/BasicLTI.xml I_847b007b-cf59-3ec8-8645-1334ff0ba478_R/BasicLTI.xml I_84845cbb-95c9-3bbc-9b0d-5f89d0fd0b9d_R/BasicLTI.xml I_848a87ce-5329-3329-a203-1db34ded6c6e_1_R/BasicLTI.xml I_848c2727-f86e-335e-9298-1f8c2bb847c3_1_R/BasicLTI.xml I_849a8627-b931-3dde-b913-78af56ec6690_1_R/BasicLTI.xml I_849c7b23-6eb7-3bff-98f8-63fe88e5d5bb_R/BasicLTI.xml I_84b856f8-6fa5-311a-8db8-493a816917da_1_R/BasicLTI.xml I_84d1e173-b6bb-3ca3-9c05-81f51ec7ee7b_1_R/BasicLTI.xml I_84d97e6b-b38c-358b-8960-d801bf9720c1_1_R/BasicLTI.xml I_84f22796-611a-3da7-9613-c4b5ddd18ce0_1_R/BasicLTI.xml I_84f22796-611a-3da7-9613-c4b5ddd18ce0_3_R/BasicLTI.xml I_85012b8d-dab7-3684-9e80-198fd0e4c033_1_R/BasicLTI.xml I_850ba648-9070-3c05-b316-344d04e55eb3_1_R/BasicLTI.xml I_85128835-759e-32bb-b7a2-ddc8a2c162d6_1_R/BasicLTI.xml I_85197282-69fd-3e1e-9a73-7dec016c9db6_1_R/BasicLTI.xml I_851c7062-c85b-3df2-b7c8-4e8194f06bc8_1_R/BasicLTI.xml I_85454c39-378b-362e-860a-598b9400252b_R/BasicLTI.xml I_854ffe0e-4590-3249-bc5f-075bf7712e84_1_R/BasicLTI.xml I_858ea6d7-2f03-33f7-970b-eb33074c9b84_1_R/BasicLTI.xml I_85adc959-cda2-337c-987a-6b015ed25b52_R/BasicLTI.xml I_85c41a68-04a2-379f-a484-c9f485bb230d_1_R/BasicLTI.xml I_85ca4c55-8b8a-3de6-9c4c-3606f08b5e7a_1_R/BasicLTI.xml I_85d96d3d-8f30-3777-a21d-dea79f86c344_1_R/BasicLTI.xml I_85f315b6-1609-37c3-941e-14ae7936e96c_1_R/BasicLTI.xml I_85fdf102-110a-3a84-858c-ca484ba7d3e8_1_R/BasicLTI.xml I_860f53f7-5cdd-3530-a17a-300d3c8182a3_R/BasicLTI.xml I_86156a47-4cb2-36e0-85e6-6e62cf42b933_1_R/BasicLTI.xml I_86156a47-4cb2-36e0-85e6-6e62cf42b933_3_R/BasicLTI.xml I_86249375-12ab-39d2-a417-21628bc3ba6a_1_R/BasicLTI.xml I_862bc06e-d92f-332c-b788-d0c619a0eccb_1_R/BasicLTI.xml I_866106a4-ba10-369d-8dc5-78f0a541cd89_R/BasicLTI.xml I_86667ba4-e7d6-3468-9598-10d91fa2aae2_R/BasicLTI.xml I_866d60f8-b685-3400-8ec4-913db0544b17_R/BasicLTI.xml I_867b6d50-2677-314b-9cec-52a62b95bce0_1_R/BasicLTI.xml I_868ea417-77bd-3b20-9a6e-1978964f4b77_R/BasicLTI.xml I_86b42feb-fc07-3df5-a6a0-1a5954bff810_R/BasicLTI.xml I_86c3756d-10f5-3d5d-8236-77f25af6f34b_1_R/BasicLTI.xml I_86c4dbd8-d3e4-37d5-9710-76e09e62fa56_1_R/BasicLTI.xml I_86fdd40b-b3b9-3514-9a6c-3b289132d760_R/BasicLTI.xml I_86ff3fcb-5714-3b74-994b-4b2475d16dd5_1_R/BasicLTI.xml I_870ac6be-c9ab-32a6-b54f-7cb3b1b1f748_1_R/BasicLTI.xml I_87181ad0-dac7-343b-9290-9d61dbb07412_1_R/BasicLTI.xml I_871b8bad-b18d-3278-b344-52e1ea9111f1_R/BasicLTI.xml I_871e898f-914b-3ce4-a296-d2b100b9d6f6_1_R/BasicLTI.xml I_8720f755-f45c-3df5-9681-26b7c7d8a81c_1_R/BasicLTI.xml I_87231c4f-5080-3c77-81c7-777eca545721_1_R/BasicLTI.xml I_87298174-3bf9-3e57-b9ea-d5e5e3cdf598_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_11_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_1_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_3_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_5_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_7_R/BasicLTI.xml I_87312674-df10-30bc-a76f-7c9dacddffff_9_R/BasicLTI.xml I_8771f5fe-31f3-3c76-95d4-bccef8c1336f_R/BasicLTI.xml I_8780c441-51cb-3749-8b34-0268a7e74ebf_1_R/BasicLTI.xml I_87b4365a-02d2-3b93-8d72-3c17254d05ba_R/BasicLTI.xml I_8833e44e-358e-3a36-96f8-01b3596a93c2_R/BasicLTI.xml I_8840a0c3-ead2-3fb9-8eea-9fb0fb4c78bb_R/BasicLTI.xml I_8842d7ad-d5c7-3df7-bb73-b410a8bf0e0a_R/BasicLTI.xml I_884d3388-6896-3a92-96ac-7be152ee62d3_1_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_11_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_13_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_15_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_1_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_3_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_5_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_7_R/BasicLTI.xml I_88550324-121e-3366-869c-015e166b8d0f_9_R/BasicLTI.xml I_88553305-ec8f-3012-8d44-6c512c9e0629_R/BasicLTI.xml I_885a3392-6f47-3115-a7b7-c1eb789072bf_R/BasicLTI.xml I_8873fce9-f573-3fb9-bb4e-f4fd73141d8a_1_R/BasicLTI.xml I_88ac8cd1-6d72-3001-816d-6fdbf4818f07_1_R/BasicLTI.xml I_88c81479-c754-3c70-b7ac-46042b6bcc23_R/BasicLTI.xml I_88d13438-b3e3-3f48-8557-02408a57212e_R/BasicLTI.xml I_89004be4-ed20-3725-878e-135176aeb891_1_R/BasicLTI.xml I_89004be4-ed20-3725-878e-135176aeb891_3_R/BasicLTI.xml I_89004be4-ed20-3725-878e-135176aeb891_5_R/BasicLTI.xml I_89004be4-ed20-3725-878e-135176aeb891_7_R/BasicLTI.xml I_89071f40-a234-3a92-a41c-21a0aebd57e6_R/BasicLTI.xml I_8912a391-2e59-36a6-ba95-f7df16559668_1_R/BasicLTI.xml I_8944d5f8-9e98-3a95-8ce0-ee3c4d37b1ba_1_R/BasicLTI.xml I_894703eb-25cc-3a84-bc78-32ca19df3ad3_1_R/BasicLTI.xml I_894703eb-25cc-3a84-bc78-32ca19df3ad3_3_R/BasicLTI.xml I_894703eb-25cc-3a84-bc78-32ca19df3ad3_5_R/BasicLTI.xml I_894703eb-25cc-3a84-bc78-32ca19df3ad3_7_R/BasicLTI.xml I_89839cb0-8a7a-3601-830e-11b125640441_1_R/BasicLTI.xml I_898af2f6-2bb2-394d-9342-3c1f5ea62d58_1_R/BasicLTI.xml I_8997beb9-0137-39fc-a140-09b40e808676_R/BasicLTI.xml I_89c82e41-66a0-378d-b8a5-f609e30768a1_1_R/BasicLTI.xml I_89c82e41-66a0-378d-b8a5-f609e30768a1_3_R/BasicLTI.xml I_89d3d986-25df-3f83-99fe-ad7472b68e0e_1_R/BasicLTI.xml I_89e2a6b5-6102-308e-ae3f-37a705a99d9a_1_R/BasicLTI.xml I_8a0f0f2d-b6d3-31e2-877e-24ebbe2ebe5b_1_R/BasicLTI.xml I_8a115be3-1180-399a-af17-f03e670cdf2f_R/BasicLTI.xml I_8a332b44-6e06-34e2-b1d5-5d10014dd9b0_1_R/BasicLTI.xml I_8a3b0542-9399-369d-b929-f2130e568ef5_1_R/BasicLTI.xml I_8a3b0542-9399-369d-b929-f2130e568ef5_3_R/BasicLTI.xml I_8a3b0542-9399-369d-b929-f2130e568ef5_5_R/BasicLTI.xml I_8a3b0542-9399-369d-b929-f2130e568ef5_7_R/BasicLTI.xml I_8a3b0542-9399-369d-b929-f2130e568ef5_9_R/BasicLTI.xml I_8a4222c0-3470-3f85-be3a-89ba8d19fc93_1_R/BasicLTI.xml I_8a75d1d4-85a1-3e00-8154-b6964df174a5_R/BasicLTI.xml I_8aa6475f-6c83-3588-b7ee-f81b59905c4d_1_R/BasicLTI.xml I_8ac81c9b-392b-3870-ba48-8256a9f354f2_R/BasicLTI.xml I_8ac9fe80-1390-364e-a31c-4bbc91cdd1fb_R/BasicLTI.xml I_8ad6a154-c436-35cc-b2f5-46aae7442bc2_R/BasicLTI.xml I_8adb4bfc-8514-34c4-bd69-ee9a83b65ecc_1_R/BasicLTI.xml I_8b0ef9c5-ce32-3160-8332-5877625b79e2_R/BasicLTI.xml I_8b109257-f015-3154-8d38-b903dc9e7389_R/BasicLTI.xml I_8b1902df-649f-35c0-88d6-b674f3f4c72d_R/BasicLTI.xml I_8b1cfc56-08df-34a7-b815-f437f72247e8_1_R/BasicLTI.xml I_8b33c42e-2bd0-380f-a659-988b9188aa25_1_R/BasicLTI.xml I_8b4de99c-2be5-3ede-bef1-9c861c462fed_1_R/BasicLTI.xml I_8b4de99c-2be5-3ede-bef1-9c861c462fed_3_R/BasicLTI.xml I_8b4e1651-c924-37b8-ab09-33502e4f6e34_1_R/BasicLTI.xml I_8b98615f-977c-3006-a86d-d7467d58eb5d_1_R/BasicLTI.xml I_8ba830dd-6964-397f-8309-92c3b84e5068_R/BasicLTI.xml I_8bc57672-f17b-32cd-934d-9ccb7d4307ae_R/BasicLTI.xml I_8bcafe88-3162-3a46-8a68-3dbf8383bf8a_1_R/BasicLTI.xml I_8be62e5f-5ecb-33bc-9363-45d4e54f1f6c_R/BasicLTI.xml I_8c062ceb-4ade-3642-8b32-2e0b8b802714_1_R/BasicLTI.xml I_8c0714a6-efc1-3be7-a5f9-b20549a3d3eb_1_R/BasicLTI.xml I_8c4277d2-8ec9-36ca-9994-cd6e82cb741d_1_R/BasicLTI.xml I_8c56323b-023b-376f-92ec-656796f01882_1_R/BasicLTI.xml I_8c5b2b38-fbca-37b5-937d-e628fab3663f_1_R/BasicLTI.xml I_8c72ca46-04f2-3279-afb0-678c829eb288_1_R/BasicLTI.xml I_8c7bb7be-e0a2-352c-83b3-0591801911d7_1_R/BasicLTI.xml I_8c9661d8-4bf7-322c-a306-8890dec3fcfb_R/BasicLTI.xml I_8cb46a61-973f-3f6c-b221-2c8923bb8703_1_R/BasicLTI.xml I_8cf4cb8f-401e-3654-909b-a1a04b17269e_1_R/BasicLTI.xml I_8cf66469-f2f4-325e-873d-93909748bd01_1_R/BasicLTI.xml I_8d19adf0-1bbb-3548-815b-6b1dc935d6a0_R/BasicLTI.xml I_8d2727a7-188a-350b-8dcc-af84075d8936_1_R/BasicLTI.xml I_8d2989f8-ca76-330c-a5fb-ef1292873887_R/BasicLTI.xml I_8d2fe167-6ac6-3904-89f3-4b0a2b52f76c_R/BasicLTI.xml I_8d5fd257-93ab-3165-ac71-2251ae7269a4_1_R/BasicLTI.xml I_8d5fd257-93ab-3165-ac71-2251ae7269a4_3_R/BasicLTI.xml I_8d63a981-d10a-3781-bd30-a09c8e5ccf14_1_R/BasicLTI.xml I_8d6510cf-24da-32d6-96a9-31fc2d00d022_R/BasicLTI.xml I_8d67e3b8-da7c-3044-93d5-58de68d80f6e_1_R/BasicLTI.xml I_8d69f4f5-13e5-3053-9789-782243557e6f_R/BasicLTI.xml I_8dcbde29-9e18-3a9c-a348-ce06d5a14bba_1_R/BasicLTI.xml I_8de4e735-7c77-3269-82b7-a512232cfc5a_R/BasicLTI.xml I_8de61833-cd73-3b09-bea3-55078bcd70ed_R/BasicLTI.xml I_8deb265a-dad2-38ef-8173-f890979e2f7a_R/BasicLTI.xml I_8e31bc41-0c01-3eaa-b603-05a9eabe44e5_1_R/BasicLTI.xml I_8e639671-6aef-3997-b486-688276ad21dc_R/BasicLTI.xml I_8e78667c-019c-3dbd-aa2e-bc6b71b3c84b_R/BasicLTI.xml I_8e7ae3b1-2ed7-3a32-933c-2eaced253b88_R/BasicLTI.xml I_8e9ab72e-908c-3d80-90b1-221e5877dae4_1_R/BasicLTI.xml I_8e9f93cd-ae34-32eb-ad12-6552f51f6106_R/BasicLTI.xml I_8ea8333a-da8d-31f9-bc00-a8e453036ce3_1_R/BasicLTI.xml I_8eac45e6-74d9-3f5e-9178-4c40fdad05e7_1_R/BasicLTI.xml I_8eee8db6-1435-30b4-b6ad-1acd1a16acc8_1_R/BasicLTI.xml I_8eee8db6-1435-30b4-b6ad-1acd1a16acc8_3_R/BasicLTI.xml I_8ef9e534-210b-3c43-99ea-389c845ada6a_1_R/BasicLTI.xml I_8ef9e534-210b-3c43-99ea-389c845ada6a_3_R/BasicLTI.xml I_8f0e6b70-b141-3920-b374-74648691757d_1_R/BasicLTI.xml I_8f0e6b70-b141-3920-b374-74648691757d_3_R/BasicLTI.xml I_8f0e6b70-b141-3920-b374-74648691757d_5_R/BasicLTI.xml I_8f1df067-31ba-3da1-be15-3cd11047c4b8_R/BasicLTI.xml I_8f22ca77-ca1b-35a5-ab61-e6fc021df524_1_R/BasicLTI.xml I_8f570d6d-0edc-3a94-ab7d-f82e4065927c_1_R/BasicLTI.xml I_8f6b35bf-c248-3c22-a50f-0734e429c0aa_1_R/BasicLTI.xml I_8f7a235c-1133-3a1b-95f1-6dbda377f840_R/BasicLTI.xml I_8f97dcc6-1e7d-35c1-b6c1-d18e317bc686_1_R/BasicLTI.xml I_8fa920bf-979f-384a-b8ed-fedbd837ec87_1_R/BasicLTI.xml I_8fb0c692-d673-30bd-8d6f-7d9984a77ab1_1_R/BasicLTI.xml I_8fe2d52b-553d-3e8d-8b44-84b55774f4fb_1_R/BasicLTI.xml I_8fe43f3a-dc28-36bc-b27b-4cd39034a39e_1_R/BasicLTI.xml I_8fe49242-4e30-3d48-bde4-66a2cd80ee61_R/BasicLTI.xml I_8ff4d069-4654-3b26-80e7-da2dcbdd664f_1_R/BasicLTI.xml I_9005ed64-42d0-3380-9b8b-620c69fc2cfe_1_R/BasicLTI.xml I_901d4603-e897-3332-aadd-bec34e2c0a1d_1_R/BasicLTI.xml I_9029b520-59a1-3f7a-9ed3-72fad2623858_1_R/BasicLTI.xml I_9029b520-59a1-3f7a-9ed3-72fad2623858_3_R/BasicLTI.xml I_902bc7d7-af17-3a1e-842b-f7e2bb73258e_1_R/BasicLTI.xml I_902d597c-7ded-318e-8e54-ed16528ee648_1_R/BasicLTI.xml I_902d597c-7ded-318e-8e54-ed16528ee648_3_R/BasicLTI.xml I_902deec1-4e2a-35e2-9ee6-33ba20998878_R/BasicLTI.xml I_90565092-7a72-3e5e-8aea-b9745f77b033_R/BasicLTI.xml I_90588d58-b3f6-3ffc-a77d-a1c93716b3cc_1_R/BasicLTI.xml I_90620a19-4266-337f-94f5-5ff7dec70b9e_R/BasicLTI.xml I_9062ec95-4211-3d9c-b487-8b0b164dc7e9_1_R/BasicLTI.xml I_90676596-f1df-396b-ae6b-ce086a9ac878_1_R/BasicLTI.xml I_90690218-3298-393e-80d6-8b430e1b58ad_1_R/BasicLTI.xml I_90856a58-4bb9-3f8c-8ee4-074e00bbe3e6_1_R/BasicLTI.xml I_90856a58-4bb9-3f8c-8ee4-074e00bbe3e6_3_R/BasicLTI.xml I_908666a7-59f2-3421-a379-4c45029239c4_R/BasicLTI.xml I_909c232f-a3d5-3eb2-a5ff-0ae3818d38d6_R/BasicLTI.xml I_90ba32ee-8905-3f4c-af39-d66a6b5712a2_1_R/BasicLTI.xml I_90bc75fe-3e90-377b-9bee-de3b30373cf5_R/BasicLTI.xml I_90e25a91-b3d1-3782-86cc-2ef427bb84fc_1_R/BasicLTI.xml I_90f37d7d-38e7-3ae7-bc93-7c2fb4ffd494_R/BasicLTI.xml I_91046775-748f-32eb-97da-4f085b1af7b1_1_R/BasicLTI.xml I_911f29e5-e3c7-34e5-90c7-844f6df3f5fd_R/BasicLTI.xml I_91317383-d823-3512-8175-10498ee6e38c_1_R/BasicLTI.xml I_9144ecea-c899-339d-adcf-e21ed5303547_1_R/BasicLTI.xml I_91474196-d863-3b7a-a107-4965129676d5_1_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_11_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_13_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_1_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_3_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_5_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_7_R/BasicLTI.xml I_91626935-65ed-3a11-8056-175c1a5ba907_9_R/BasicLTI.xml I_91716a8c-93a2-3259-a786-824e4923e707_R/BasicLTI.xml I_917c6543-e449-3e11-86aa-36f43aaaf26f_R/BasicLTI.xml I_91844293-e499-3d03-9ecd-45466e7f68d9_1_R/BasicLTI.xml I_918a96a8-500a-31cc-ad2b-5bb6aa44da80_1_R/BasicLTI.xml I_919bcdc6-592c-367b-b5a9-97387e26d924_R/BasicLTI.xml I_91a1fc9d-ce35-3da2-973c-5ed26ad2d067_1_R/BasicLTI.xml I_91a57a33-e368-34b3-b324-730fb1268d75_1_R/BasicLTI.xml I_91b97555-8c41-3cac-bb70-cdc858cbee35_R/BasicLTI.xml I_91c4edf2-a7d2-3292-9c82-4a6e88d1da82_R/BasicLTI.xml I_91db922b-8dda-323e-95d9-719724f3eed8_R/BasicLTI.xml I_91ea9334-caca-39df-bb21-e3d72ced037f_1_R/BasicLTI.xml I_91ec58f1-f76f-3421-91e8-c77b948525bb_1_R/BasicLTI.xml I_92038338-5677-3d06-86eb-88e957445524_1_R/BasicLTI.xml I_92041824-fe5b-3800-bac5-bfe540a23f90_1_R/BasicLTI.xml I_92041824-fe5b-3800-bac5-bfe540a23f90_3_R/BasicLTI.xml I_92041824-fe5b-3800-bac5-bfe540a23f90_5_R/BasicLTI.xml I_92041824-fe5b-3800-bac5-bfe540a23f90_7_R/BasicLTI.xml I_922a56dc-6b19-3b91-a679-3a1fd898b805_R/BasicLTI.xml I_922e6b32-ac7c-3b3a-9e5f-a3e0f06be6ac_R/BasicLTI.xml I_923532d4-7216-3c1d-818f-9339747bdb7d_R/BasicLTI.xml I_923b1c23-8d5c-3ab7-b496-af952d973faf_R/BasicLTI.xml I_924a1f17-fc5b-310c-a642-3e9e4b818c15_1_R/BasicLTI.xml I_927a5101-a802-3ffe-8b9c-0ccd78f9677e_R/BasicLTI.xml I_927ab7a1-e88f-3db2-be51-1a180b56f8b6_1_R/BasicLTI.xml I_9296bdad-81a5-3f25-95ef-0ba22c88ad54_1_R/BasicLTI.xml I_92d31702-2e33-3648-bad6-e26e4d51cb91_1_R/BasicLTI.xml I_92ddcfa3-dd8d-3834-90a7-8bf781726b15_1_R/BasicLTI.xml I_92de8216-192e-3b05-8cf9-de2ce615ee08_1_R/BasicLTI.xml I_92de8216-192e-3b05-8cf9-de2ce615ee08_3_R/BasicLTI.xml I_92e0359c-59ec-3afc-a562-bc6521a9cabb_1_R/BasicLTI.xml I_92ec1910-dfc5-3385-be01-4e97b85e9de2_1_R/BasicLTI.xml I_92ec7d01-dc13-3e9a-853e-153c98ac65f6_1_R/BasicLTI.xml I_930257c7-565d-3b46-b1e4-2cc824838332_1_R/BasicLTI.xml I_9308e880-1445-3290-97a5-ad6f5fa7509f_1_R/BasicLTI.xml I_933863b7-ad9d-3f1c-83e7-d4621c8217ba_1_R/BasicLTI.xml I_9341fcb8-a413-30cb-9158-d39cff794292_1_R/BasicLTI.xml I_9341fcb8-a413-30cb-9158-d39cff794292_3_R/BasicLTI.xml I_9341fcb8-a413-30cb-9158-d39cff794292_5_R/BasicLTI.xml I_9341fcb8-a413-30cb-9158-d39cff794292_7_R/BasicLTI.xml I_934e3971-93cb-3c01-bb34-e48f6dfad958_1_R/BasicLTI.xml I_9385632c-b8b3-3ab2-8592-cbe092f51337_1_R/BasicLTI.xml I_93b1a403-40b8-36b9-a005-027b8b4a7015_1_R/BasicLTI.xml I_93d19720-b9a2-3338-9800-08e2eafde93c_1_R/BasicLTI.xml I_93d300ff-113b-3bad-b51a-db2e9898b32a_1_R/BasicLTI.xml I_93f672f0-16dd-3cf3-b71e-b30537fe5e3a_1_R/BasicLTI.xml I_93f672f0-16dd-3cf3-b71e-b30537fe5e3a_3_R/BasicLTI.xml I_93f672f0-16dd-3cf3-b71e-b30537fe5e3a_5_R/BasicLTI.xml I_93f672f0-16dd-3cf3-b71e-b30537fe5e3a_7_R/BasicLTI.xml I_94129858-3c72-3a82-a59a-8ed9e621f370_1_R/BasicLTI.xml I_94162975-c26f-38c3-b758-4cf6628cd9cc_1_R/BasicLTI.xml I_94338f29-505e-3c02-bb6e-c5ee7217ad11_R/BasicLTI.xml I_9460d9fa-cc80-3143-b9db-651f20e60c62_R/BasicLTI.xml I_9467a69b-0dff-37df-b1da-693cd62b4b22_1_R/BasicLTI.xml I_9467a69b-0dff-37df-b1da-693cd62b4b22_3_R/BasicLTI.xml I_9471f06d-f00e-34ee-a7f7-baa7cf7b4f50_1_R/BasicLTI.xml I_94895b00-4260-37bd-b61e-318a0233c145_R/BasicLTI.xml I_94d11613-1f30-364a-8e63-a20226ecbfd7_R/BasicLTI.xml I_94d186ab-191c-3daf-bca8-3ea17561eca8_1_R/BasicLTI.xml I_94dc9f12-3761-3dc3-a38c-9da648b937b1_R/BasicLTI.xml I_94fa3372-8909-3b17-9d22-6e38e57bebc2_1_R/BasicLTI.xml I_953e8237-d544-3668-911f-2ba75f9b163f_R/BasicLTI.xml I_95490191-34ad-3b95-a6c6-f05b39746a83_R/BasicLTI.xml I_9554c6b5-8fa6-38e8-b22b-3a02192cc73d_1_R/BasicLTI.xml I_9554c6b5-8fa6-38e8-b22b-3a02192cc73d_3_R/BasicLTI.xml I_956da23c-98a2-373d-904f-18e0df4e10f8_1_R/BasicLTI.xml I_957e512e-333f-3df3-b014-20bfc5000cb2_R/BasicLTI.xml I_95800650-1cb4-3915-9d9b-afb75f4a7ce5_1_R/BasicLTI.xml I_95824c5b-dd10-31f3-94bd-3759bf31640a_1_R/BasicLTI.xml I_95824c5b-dd10-31f3-94bd-3759bf31640a_3_R/BasicLTI.xml I_958938b5-16cc-3b12-82cd-66256e43ac1e_1_R/BasicLTI.xml I_958938b5-16cc-3b12-82cd-66256e43ac1e_3_R/BasicLTI.xml I_958938b5-16cc-3b12-82cd-66256e43ac1e_5_R/BasicLTI.xml I_958f2b33-1bd9-3ace-8f51-b96190a912ad_1_R/BasicLTI.xml I_959e91a6-f127-3525-a986-5ba19d93b6e8_1_R/BasicLTI.xml I_95a19580-7ce8-3660-833d-73ec12f4b336_1_R/BasicLTI.xml I_95b90a54-7d86-3e23-bfe9-4a22427d3b42_R/BasicLTI.xml I_95f696d5-8c9f-30c0-8d33-3006d8e8c3d3_R/BasicLTI.xml I_96273ffc-5f53-3f43-b5c2-57017062f8b7_1_R/BasicLTI.xml I_96296f04-6e10-3a1f-bc26-66dcc41ff18b_R/BasicLTI.xml I_9659173b-0c0d-3641-ba83-fb0345c6dd9f_1_R/BasicLTI.xml I_967d232c-bb63-3692-b4d4-9591b4779ac2_R/BasicLTI.xml I_9696f17a-db46-3af9-9631-4dd127bd8751_1_R/BasicLTI.xml I_96a76162-1477-3434-88ff-312b9f2801f8_1_R/BasicLTI.xml I_96a76162-1477-3434-88ff-312b9f2801f8_3_R/BasicLTI.xml I_96b4c3f2-1fc0-3670-a108-666db37aeea9_1_R/BasicLTI.xml I_96d313f8-5527-3982-9523-ccf80cdbd39c_1_R/BasicLTI.xml I_96f73abc-b3b9-3971-9a34-4c4e3a0cc8d6_1_R/BasicLTI.xml I_96f986fa-3062-343d-bc54-e284556a8d50_1_R/BasicLTI.xml I_96ffa2ec-6031-3988-9f7b-43505dbc1c68_R/BasicLTI.xml I_9728f385-c236-3261-a0f5-7f9c87909a3a_1_R/BasicLTI.xml I_9730475e-b669-3800-aef1-9b476d0c2e7e_R/BasicLTI.xml I_974910fc-2108-38ba-9a80-32d76151b27b_1_R/BasicLTI.xml I_9794d27a-848c-39a3-8add-e8bc85ed9e70_1_R/BasicLTI.xml I_979b8d90-1ee5-3b24-b860-d41e0d04b318_R/BasicLTI.xml I_97b729db-9e93-361f-9ed6-7ae6083cc32b_R/BasicLTI.xml I_97b8ec60-7ab2-37db-b8bd-cd318c20ee66_1_R/BasicLTI.xml I_97c634fc-eb0b-3f65-b67b-58567401d3eb_1_R/BasicLTI.xml I_97cc1473-069a-3a9c-997d-64eb702ecd8b_1_R/BasicLTI.xml I_97d210d4-86f0-365a-87d3-2df8e6668be2_R/BasicLTI.xml I_97d51493-6725-39d0-a561-aaeac982051c_1_R/BasicLTI.xml I_97d9d4c9-c305-3549-b9d6-68e14b4f4555_1_R/BasicLTI.xml I_97f88eb5-156c-3868-a4c8-45d44735bf33_1_R/BasicLTI.xml I_97fb9bc7-9312-34b2-a2d0-f196c65111a2_1_R/BasicLTI.xml I_981a473c-dd63-3900-b9d4-4d6769e12e3a_1_R/BasicLTI.xml I_981a473c-dd63-3900-b9d4-4d6769e12e3a_3_R/BasicLTI.xml I_9837ab32-b93c-3c8a-92da-06c6f66a00d6_R/BasicLTI.xml I_98617be2-fea7-3b0a-8281-bf866f55bf94_1_R/BasicLTI.xml I_988b3fed-4bac-3a91-9001-b96a5424d42a_1_R/BasicLTI.xml I_98a30efd-a7b5-3be7-8acf-b0f43ef4ef98_1_R/BasicLTI.xml I_98a94db9-9c54-32c8-b993-dbfeaa6edb50_1_R/BasicLTI.xml I_98bd9a40-3ec2-3748-ac0c-b6f7a838b357_1_R/BasicLTI.xml I_98c6b1cd-6335-30b6-a3be-c1406ced2be6_1_R/BasicLTI.xml I_98c6b1cd-6335-30b6-a3be-c1406ced2be6_3_R/BasicLTI.xml I_98c6b1cd-6335-30b6-a3be-c1406ced2be6_5_R/BasicLTI.xml I_98f96b52-8026-3b18-86b0-5cb6ad238a17_R/BasicLTI.xml I_991f7900-b445-3e8a-b31a-828784533bcb_1_R/BasicLTI.xml I_993400e9-782d-3d6a-8529-754414e4fdcf_R/BasicLTI.xml I_99433e8c-a026-3dcc-9f68-2593dfbc4692_R/BasicLTI.xml I_994793fe-3b34-326b-a1ee-ee5e4fdbb571_1_R/BasicLTI.xml I_99510d06-6eeb-30f7-8794-ea7e364ed7e2_1_R/BasicLTI.xml I_996f0fd0-aaf4-3c98-848d-ab056ac7d7ba_1_R/BasicLTI.xml I_999febb7-d2a7-3d35-8f48-329ff2ed849d_1_R/BasicLTI.xml I_99bb6284-be9e-3911-9231-cb9c12232561_1_R/BasicLTI.xml I_99e56dae-b7c3-36b6-96ab-ee7556aa07b9_R/BasicLTI.xml I_99efd70f-f5b7-3c40-ab2a-a34ce304635e_1_R/BasicLTI.xml I_99fd57a2-aa0e-3d38-9486-372c7aa85413_1_R/BasicLTI.xml I_9a029eab-9848-3006-b219-773dda8d3e1b_R/BasicLTI.xml I_9a143db2-3050-3393-85b4-f8cd9986f80f_1_R/BasicLTI.xml I_9a26782f-5911-3a01-87a7-3c5dfaf94056_R/BasicLTI.xml I_9a277b05-390f-3481-b25d-328c7f9a5ef9_1_R/BasicLTI.xml I_9a432210-9194-3a09-96a8-08e1d4f82733_1_R/BasicLTI.xml I_9a483670-3a97-3a8c-9038-5e1c63a0cefa_1_R/BasicLTI.xml I_9a4cbecb-4cc5-38d6-8da7-fcc3bfe6813e_1_R/BasicLTI.xml I_9a50ad58-dd57-3351-8020-13469f6ec5d5_R/BasicLTI.xml I_9a89f69e-90fb-3444-b8fe-4d618f9843c2_1_R/BasicLTI.xml I_9a8d0b8f-4169-3dbd-8f42-ec63e9461309_R/BasicLTI.xml I_9a9b06a4-cacd-3587-adb3-6861b8d845bb_R/BasicLTI.xml I_9ab2bbdc-5b33-3ac0-bdc8-c0410568c9ff_R/BasicLTI.xml I_9ae29275-2e44-3fa1-b299-23b20386131b_1_R/BasicLTI.xml I_9af21dd9-956d-3fff-ba8b-389a0a161646_1_R/BasicLTI.xml I_9af21dd9-956d-3fff-ba8b-389a0a161646_3_R/BasicLTI.xml I_9af56ec8-13c0-304e-856c-8e921631dee7_R/BasicLTI.xml I_9b2783c8-27a0-3b48-9a05-0a7ea1822d8e_1_R/BasicLTI.xml I_9b2783c8-27a0-3b48-9a05-0a7ea1822d8e_3_R/BasicLTI.xml I_9b2783c8-27a0-3b48-9a05-0a7ea1822d8e_5_R/BasicLTI.xml I_9b2783c8-27a0-3b48-9a05-0a7ea1822d8e_7_R/BasicLTI.xml I_9b2ab964-526a-3144-bc01-d510d9644bb5_1_R/BasicLTI.xml I_9b3d2f8b-5b98-340c-ab3a-98eed8247a23_R/BasicLTI.xml I_9b3d324c-d884-3650-abde-ba2dac53a4ba_1_R/BasicLTI.xml I_9b5a2ceb-8bc7-3e78-becf-7595acb73912_R/BasicLTI.xml I_9b7eb2d5-e465-3bc2-bbf3-d913fe8f3b95_1_R/BasicLTI.xml I_9b847e73-3a09-31e7-a606-b39477a0d7c3_1_R/BasicLTI.xml I_9b97e309-6560-3ad5-b87d-343c7b08240d_1_R/BasicLTI.xml I_9b97e309-6560-3ad5-b87d-343c7b08240d_3_R/BasicLTI.xml I_9baf6416-a74f-33ad-a647-26ce89602b55_1_R/BasicLTI.xml I_9bcbc1d8-8e39-331e-9d59-e64d6750d08f_1_R/BasicLTI.xml I_9bdcaf66-9c8d-3adf-81f2-ee0679663f4f_R/BasicLTI.xml I_9bf13092-f438-3a5b-bd74-86db02353d4e_1_R/BasicLTI.xml I_9c06f02b-0c05-310e-be2b-4f731a205498_1_R/BasicLTI.xml I_9c0c072b-0bdb-3a2a-83b3-d4b33dd77990_R/BasicLTI.xml I_9c36a800-6c41-383d-9e37-4619aa8979a6_R/BasicLTI.xml I_9c4271bf-d4fb-3b0d-b96a-bb0eb6d48762_1_R/BasicLTI.xml I_9c4271bf-d4fb-3b0d-b96a-bb0eb6d48762_3_R/BasicLTI.xml I_9c527ad8-2db0-3a3b-8fc0-f9aac17c2645_1_R/BasicLTI.xml I_9c6ce1eb-7051-3bbd-ba85-0e7dc5eb2db9_1_R/BasicLTI.xml I_9c86d901-0f53-3a8e-bf98-b822ea91f409_1_R/BasicLTI.xml I_9c9659d1-0754-3e07-93cc-d13615fd1c7b_R/BasicLTI.xml I_9c9f7965-bf1d-37ba-9e26-9091aabfcda5_1_R/BasicLTI.xml I_9ca1d156-d8c9-30ca-a0f0-020350708338_1_R/BasicLTI.xml I_9ca26058-f841-3dc7-9fed-b058c5d8b809_1_R/BasicLTI.xml I_9ca26058-f841-3dc7-9fed-b058c5d8b809_3_R/BasicLTI.xml I_9ca26058-f841-3dc7-9fed-b058c5d8b809_5_R/BasicLTI.xml I_9ca26058-f841-3dc7-9fed-b058c5d8b809_7_R/BasicLTI.xml I_9cc4f477-d430-3f15-bad9-d29944d720d2_1_R/BasicLTI.xml I_9ce3944e-c8c9-3e02-aadb-5edb979a1b44_1_R/BasicLTI.xml I_9ce7250d-7179-3cf8-9a8c-5e1d9050f4f4_R/BasicLTI.xml I_9ce7a303-d126-36a5-9247-36de2ca63eba_1_R/BasicLTI.xml I_9ce999f1-0ec3-3c1a-8598-3fb84c8719cb_R/BasicLTI.xml I_9cf35d51-184e-3828-9fb5-379e94aff022_1_R/BasicLTI.xml I_9cf35d51-184e-3828-9fb5-379e94aff022_3_R/BasicLTI.xml I_9cf35d51-184e-3828-9fb5-379e94aff022_5_R/BasicLTI.xml I_9cf35d51-184e-3828-9fb5-379e94aff022_7_R/BasicLTI.xml I_9cf35d51-184e-3828-9fb5-379e94aff022_9_R/BasicLTI.xml I_9cff8f19-851d-370e-ac4f-9e471b093a2d_1_R/BasicLTI.xml I_9d09c2ce-17f3-378d-87f6-2f92229043ac_1_R/BasicLTI.xml I_9d0ae458-2d93-3deb-a30e-2dff16ff36a3_1_R/BasicLTI.xml I_9d0ae458-2d93-3deb-a30e-2dff16ff36a3_3_R/BasicLTI.xml I_9d26e69e-053a-3375-ab9f-d8d2eabe74ca_1_R/BasicLTI.xml I_9d2bfbc3-e46d-3e3d-89fb-e137d1d437a7_R/BasicLTI.xml I_9d368ca6-634d-3843-bdb4-c376d045d8be_1_R/BasicLTI.xml I_9d368ca6-634d-3843-bdb4-c376d045d8be_3_R/BasicLTI.xml I_9d38b88f-e99f-3821-affc-4f4557691f4e_R/BasicLTI.xml I_9d3b5e48-f777-39ac-b8bc-3b36cfd1d8f4_1_R/BasicLTI.xml I_9d424d90-961c-3535-b560-eaec070ae538_1_R/BasicLTI.xml I_9d4ae0e7-189d-3430-a91c-14ca8d541952_R/BasicLTI.xml I_9d57587f-cd19-3331-8870-4df3f234e871_R/BasicLTI.xml I_9d7ad709-7ee1-3907-980f-bc3ffb6270b1_1_R/BasicLTI.xml I_9d7ad709-7ee1-3907-980f-bc3ffb6270b1_3_R/BasicLTI.xml I_9d7ad709-7ee1-3907-980f-bc3ffb6270b1_5_R/BasicLTI.xml I_9d7da50d-f0e1-3893-88ff-18d40e38058b_1_R/BasicLTI.xml I_9d7f8383-5e9a-3123-b5bf-3c762c281040_1_R/BasicLTI.xml I_9d822c5b-2e8b-3542-a8f5-30980e47721f_R/BasicLTI.xml I_9d83efbf-c383-38e8-86c5-c46dfe0a997c_R/BasicLTI.xml I_9da53ef6-c682-3eff-a25b-34f8d49576fb_R/BasicLTI.xml I_9dc6cfba-738d-36ce-a696-79164a6d4f57_1_R/BasicLTI.xml I_9dc7a566-9bab-3f34-854c-d4a81c997b15_1_R/BasicLTI.xml I_9dc7a566-9bab-3f34-854c-d4a81c997b15_3_R/BasicLTI.xml I_9dde58f8-5dc3-39a8-99e2-fec843035e02_R/BasicLTI.xml I_9de11166-7ea5-3c95-8874-4f2a8dad5493_1_R/BasicLTI.xml I_9de209dc-d788-3f84-94c3-62327e782c14_1_R/BasicLTI.xml I_9e00ad7c-818b-35ee-bc45-587c223c42c4_R/BasicLTI.xml I_9e18eb0b-d1f3-3a9d-9af2-3f0837178c5f_1_R/BasicLTI.xml I_9e4dadc6-3348-3b8a-8d3a-35ec4adc3196_R/BasicLTI.xml I_9e4e09bf-eea0-38bd-a077-00ed01ddecac_1_R/BasicLTI.xml I_9e4e09bf-eea0-38bd-a077-00ed01ddecac_3_R/BasicLTI.xml I_9e4e93da-0d30-327e-80f3-e63f4bd080d5_R/BasicLTI.xml I_9e4ebd6c-4336-3d5c-b44b-4560fc669d40_1_R/BasicLTI.xml I_9e4ebd6c-4336-3d5c-b44b-4560fc669d40_3_R/BasicLTI.xml I_9e4ebd6c-4336-3d5c-b44b-4560fc669d40_5_R/BasicLTI.xml I_9e4ebd6c-4336-3d5c-b44b-4560fc669d40_7_R/BasicLTI.xml I_9e4ebd6c-4336-3d5c-b44b-4560fc669d40_9_R/BasicLTI.xml I_9e6dbf5e-a9a1-39c7-ac36-914e0c1f572b_1_R/BasicLTI.xml I_9e772260-6349-3083-bab4-ddddaa1650ce_1_R/BasicLTI.xml I_9e8233ee-076d-3156-af6c-a9e745bb26f9_R/BasicLTI.xml I_9e8c9eac-b2fe-3be1-8e00-8c76bc589c32_R/BasicLTI.xml I_9ea7b7a2-4a4d-3e86-9a57-6e2c93f744ae_R/BasicLTI.xml I_9edde912-a2f5-347d-b219-16d3ca9abd50_1_R/BasicLTI.xml I_9ef0f936-b5d5-34f8-bf3f-a8b3197346ea_R/BasicLTI.xml I_9efd29da-9325-33df-a620-634f54e2d138_R/BasicLTI.xml I_9f0220c0-d410-3914-a09d-5f03c8649721_1_R/BasicLTI.xml I_9f154d51-603d-3f64-8da1-1a45dd732b1d_R/BasicLTI.xml I_9f222af7-f1a1-3477-8062-9e3958546900_1_R/BasicLTI.xml I_9f40c51b-845c-382d-839b-7a1e428abe8c_1_R/BasicLTI.xml I_9f689314-4ad2-37d6-a687-3e90a4d6185c_1_R/BasicLTI.xml I_9f920300-257d-31eb-83e9-1bc434423e86_R/BasicLTI.xml I_9f921184-88b9-3d22-9247-b07bb51e682a_1_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_11_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_1_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_3_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_5_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_7_R/BasicLTI.xml I_9f97e308-6c0b-3ce3-bc51-d9d0ff5d3035_9_R/BasicLTI.xml I_9fb02a7c-e977-3390-a2cd-890a09158b22_1_R/BasicLTI.xml I_9fb02a7c-e977-3390-a2cd-890a09158b22_3_R/BasicLTI.xml I_9fd19b8f-97e2-3407-bbdc-f70b9b0e3728_1_R/BasicLTI.xml I_9ff89502-768e-37c6-85af-80c565e2801d_R/BasicLTI.xml I_9ffba169-835b-308b-8911-66a94602c990_R/BasicLTI.xml I_9ffcc35c-a7c8-390e-9199-f337e6ddda40_1_R/BasicLTI.xml I_a00c8854-bcbe-3b74-b9df-d3f120ccb9e5_1_R/BasicLTI.xml I_a011f567-3e56-3f2d-84ed-fb85d11c4b59_1_R/BasicLTI.xml I_a03348c4-65f4-3867-a25d-ddbc1e7c1e2c_1_R/BasicLTI.xml I_a059a514-49d6-3c89-a49a-44c208cdde0f_1_R/BasicLTI.xml I_a05b00bf-4ab0-3ca0-a411-7f5b6f75ee1a_R/BasicLTI.xml I_a070b04e-21af-36f6-bd8c-1479575f5294_1_R/BasicLTI.xml I_a077b8ff-1b70-3afa-b857-f6ccd2b77102_R/BasicLTI.xml I_a07ebb3a-dc49-3ea5-a1c0-cdb8c12099ac_1_R/BasicLTI.xml I_a088f269-4924-3d21-b93c-ac154af3053d_R/BasicLTI.xml I_a0919e8c-74bc-38a7-b1bd-8adee3fe5b51_R/BasicLTI.xml I_a09eb8a5-eb07-354c-8368-c55948731f35_1_R/BasicLTI.xml I_a09eb8a5-eb07-354c-8368-c55948731f35_3_R/BasicLTI.xml I_a0a116d6-55c0-306c-ba43-402e52076dc5_1_R/BasicLTI.xml I_a0dc47bd-4f93-319c-8a08-038a5bcb7f27_R/BasicLTI.xml I_a10f2d31-aaf8-3ea2-aee8-cc57343b7458_1_R/BasicLTI.xml I_a110468d-87a8-3be5-8b08-07f502f40639_1_R/BasicLTI.xml I_a13def48-0484-3672-9896-1c3b0314875f_1_R/BasicLTI.xml I_a13feca0-9484-3800-ad19-c1dd9af4eeab_1_R/BasicLTI.xml I_a143280f-0323-3709-a189-8e3337559fac_1_R/BasicLTI.xml I_a17d002c-7522-30f8-b4aa-2c0dbbc5b1c5_1_R/BasicLTI.xml I_a185bd07-8e13-32fc-aaab-041f03787992_1_R/BasicLTI.xml I_a19259ed-1b02-314c-87a1-331583089012_1_R/BasicLTI.xml I_a19f38bf-4a0a-3427-a6f2-f2db63de3276_R/BasicLTI.xml I_a1bd862f-a5e2-31a1-84c7-98db53afe74e_1_R/BasicLTI.xml I_a1bd862f-a5e2-31a1-84c7-98db53afe74e_3_R/BasicLTI.xml I_a1df94de-f239-3813-ab3c-6fa287e13af6_1_R/BasicLTI.xml I_a1df94de-f239-3813-ab3c-6fa287e13af6_3_R/BasicLTI.xml I_a1df94de-f239-3813-ab3c-6fa287e13af6_5_R/BasicLTI.xml I_a1ebad46-d420-32d5-9562-421afe1aa4cf_R/BasicLTI.xml I_a204c922-e37b-3c3d-bff7-25638a81a878_1_R/BasicLTI.xml I_a204c922-e37b-3c3d-bff7-25638a81a878_3_R/BasicLTI.xml I_a204c922-e37b-3c3d-bff7-25638a81a878_5_R/BasicLTI.xml I_a204c922-e37b-3c3d-bff7-25638a81a878_7_R/BasicLTI.xml I_a20d5b19-202d-39bf-8cec-2510e7ffe852_1_R/BasicLTI.xml I_a20d5b19-202d-39bf-8cec-2510e7ffe852_3_R/BasicLTI.xml I_a21478a9-562c-3031-97e5-8ac74ece88d1_1_R/BasicLTI.xml I_a22ab9eb-0191-365f-bb4d-f9c107df5137_R/BasicLTI.xml I_a22bb05f-ae3e-3101-8668-742feba8a79a_R/BasicLTI.xml I_a23e6dba-2f87-3a19-9052-3298c600ea79_R/BasicLTI.xml I_a2628501-539f-38e8-84a4-c2c94ee75e3b_R/BasicLTI.xml I_a289bf1a-9d5b-3622-b4b6-587fe67104e7_R/BasicLTI.xml I_a28dff04-4923-36b3-beaf-60120a6a69ee_1_R/BasicLTI.xml I_a2a0606c-e95a-33b2-9abc-bf0aeee94fa9_1_R/BasicLTI.xml I_a2b7f278-a380-3f91-aa29-5eee1a4f0a28_R/BasicLTI.xml I_a2b919ff-88a0-3deb-bec7-dcd16eff3cd4_1_R/BasicLTI.xml I_a2e9a223-366d-3867-9a58-05ae3c5b0362_R/BasicLTI.xml I_a2fc794c-96f8-3f84-9b07-d9c14eb3e223_1_R/BasicLTI.xml I_a2fc794c-96f8-3f84-9b07-d9c14eb3e223_3_R/BasicLTI.xml I_a2fc794c-96f8-3f84-9b07-d9c14eb3e223_5_R/BasicLTI.xml I_a307b9f4-d45e-3d31-b896-c179172efbcc_1_R/BasicLTI.xml I_a307b9f4-d45e-3d31-b896-c179172efbcc_3_R/BasicLTI.xml I_a307b9f4-d45e-3d31-b896-c179172efbcc_5_R/BasicLTI.xml I_a307b9f4-d45e-3d31-b896-c179172efbcc_7_R/BasicLTI.xml I_a307b9f4-d45e-3d31-b896-c179172efbcc_9_R/BasicLTI.xml I_a336fa20-a23c-3e78-bf54-8cf0b8924856_1_R/BasicLTI.xml I_a34796c6-7985-3e68-9e26-fc608d6f6e33_R/BasicLTI.xml I_a35b94ab-5a00-3a90-93b5-beeb81cf73f4_1_R/BasicLTI.xml I_a35ec56c-ce20-36c3-9d48-ffdcf28d0c70_R/BasicLTI.xml I_a3611a32-712e-321c-97a9-128d08087088_1_R/BasicLTI.xml I_a3611a32-712e-321c-97a9-128d08087088_3_R/BasicLTI.xml I_a3611a32-712e-321c-97a9-128d08087088_5_R/BasicLTI.xml I_a3611a32-712e-321c-97a9-128d08087088_7_R/BasicLTI.xml I_a3611a32-712e-321c-97a9-128d08087088_9_R/BasicLTI.xml I_a365dd4f-bf35-38db-9179-fe7e5bb15706_1_R/BasicLTI.xml I_a365dd4f-bf35-38db-9179-fe7e5bb15706_3_R/BasicLTI.xml I_a3be9851-2fec-3e19-8db2-9769a38a8925_1_R/BasicLTI.xml I_a3c85b11-e70b-3a02-a798-252467b7b395_R/BasicLTI.xml I_a3d7bd46-5534-306b-9273-1e09e678809b_1_R/BasicLTI.xml I_a3da4108-6bc4-33a2-8a72-5fc9611be43c_R/BasicLTI.xml I_a3dbab68-22eb-39da-a9fa-11800d8a2b1e_1_R/BasicLTI.xml I_a3e317d9-e73b-3c1c-a17d-e15d59092bad_1_R/BasicLTI.xml I_a3eab5fb-087e-3123-93dd-26e1c455eda0_R/BasicLTI.xml I_a3eef9d3-0107-38bd-9782-1c7aaec27b82_1_R/BasicLTI.xml I_a3fbfa7f-e086-3e33-843f-3277ca8e5d9b_1_R/BasicLTI.xml I_a410e3f7-2b86-384e-ab30-5dc3c3e6fc4f_R/BasicLTI.xml I_a41ebb96-7d0e-3e9b-90c5-405a0881bc84_R/BasicLTI.xml I_a41f576e-1bf8-3a9b-9bef-947000042892_1_R/BasicLTI.xml I_a41f576e-1bf8-3a9b-9bef-947000042892_3_R/BasicLTI.xml I_a4338ee8-7f68-3be4-b505-d533fc97c2d5_1_R/BasicLTI.xml I_a44bb3e6-5d17-33ba-aebe-d2438ba5089c_1_R/BasicLTI.xml I_a44cc73d-a741-3e18-a426-4fbbf1578169_R/BasicLTI.xml I_a46ddce6-64ca-3775-9df4-fc5e5dec5d93_R/BasicLTI.xml I_a4769598-08db-3b73-8049-f2458e563fd2_R/BasicLTI.xml I_a4841324-a2ff-3654-8287-a29c0bc37b29_R/BasicLTI.xml I_a493126c-8eca-3225-b9f6-0be7b13e2528_1_R/BasicLTI.xml I_a493126c-8eca-3225-b9f6-0be7b13e2528_3_R/BasicLTI.xml I_a4959be3-89dc-3609-a782-43291452f3e3_R/BasicLTI.xml I_a49a32e3-fbb1-3268-a1b8-f005238d9292_1_R/BasicLTI.xml I_a4c94b3c-2327-396d-9b56-09d4430e6148_1_R/BasicLTI.xml I_a4dc495c-031d-3be2-afb2-5e924342cb1c_1_R/BasicLTI.xml I_a4e41efd-a484-3a50-9b4f-a165babdaf47_1_R/BasicLTI.xml I_a4e49ad4-950b-3431-896f-f98e0b6c31ab_1_R/BasicLTI.xml I_a4e9ea4a-b040-3e0f-8765-97fa2c32917b_1_R/BasicLTI.xml I_a4e9ea4a-b040-3e0f-8765-97fa2c32917b_3_R/BasicLTI.xml I_a4eab2be-5181-3ae9-8cba-e703d45f1ab7_1_R/BasicLTI.xml I_a4eaefef-7c3b-3ece-bdd5-2f0f265a7ffb_1_R/BasicLTI.xml I_a5127025-bf0a-348b-87a6-5d50fefe28e9_R/BasicLTI.xml I_a52918fb-bbd3-31b5-b127-0f84e4dce086_1_R/BasicLTI.xml I_a55305e0-aff2-35c6-81e5-207d5b76e8f3_1_R/BasicLTI.xml I_a55305e0-aff2-35c6-81e5-207d5b76e8f3_3_R/BasicLTI.xml I_a55305e0-aff2-35c6-81e5-207d5b76e8f3_5_R/BasicLTI.xml I_a55305e0-aff2-35c6-81e5-207d5b76e8f3_7_R/BasicLTI.xml I_a55305e0-aff2-35c6-81e5-207d5b76e8f3_9_R/BasicLTI.xml I_a5536a8a-5dec-34c7-8cb4-f240caf9db2f_R/BasicLTI.xml I_a5575a64-6a2d-3c20-9cf8-e3fc44431f73_1_R/BasicLTI.xml I_a5beb6a1-dfe7-36a4-9e7c-d6f2276190c4_R/BasicLTI.xml I_a5d1ae17-654a-385e-bc6f-05d9ed2e8d86_1_R/BasicLTI.xml I_a5eddb12-0669-3bc5-90d6-02691998ca31_1_R/BasicLTI.xml I_a5f1b847-c973-3f3b-bd3d-52c1f8688329_1_R/BasicLTI.xml I_a5fd9606-976b-357b-8018-f2d6d6002f17_1_R/BasicLTI.xml I_a608f7b0-d7c5-3110-ba76-6362964b2d24_1_R/BasicLTI.xml I_a6167359-38e2-3abb-94e1-cf2ce04f6a85_1_R/BasicLTI.xml I_a61bde2d-a6cc-33e6-a34c-420825ee5f31_1_R/BasicLTI.xml I_a640cdc7-1755-32cc-8ed7-9c6fb14f016c_1_R/BasicLTI.xml I_a6472110-c82c-36ad-aa5d-8635558a7339_1_R/BasicLTI.xml I_a66aed09-6e1d-3fe0-b9b5-7bda2cff6fff_1_R/BasicLTI.xml I_a6724e91-ae6b-3621-8691-e5b8c9bff8b5_1_R/BasicLTI.xml I_a6724e91-ae6b-3621-8691-e5b8c9bff8b5_3_R/BasicLTI.xml I_a6724e91-ae6b-3621-8691-e5b8c9bff8b5_5_R/BasicLTI.xml I_a6724e91-ae6b-3621-8691-e5b8c9bff8b5_7_R/BasicLTI.xml I_a683f8e0-b925-3de1-ab82-c5d039d27c9d_1_R/BasicLTI.xml I_a68e985d-f437-3bf2-a2e3-b7f88aa7902e_1_R/BasicLTI.xml I_a6a4a634-317f-3909-aa9b-ec3605db0470_R/BasicLTI.xml I_a6b278a6-6c0a-38ea-bcae-5c8854099aa0_R/BasicLTI.xml I_a6b37d65-76f4-3aa3-9e66-89508984034f_1_R/BasicLTI.xml I_a6ead600-a76f-3230-89a6-b9963170ff1a_1_R/BasicLTI.xml I_a7007052-d221-3d81-9cc8-fcbadf23fc7e_1_R/BasicLTI.xml I_a70fd6c6-2a39-384c-8475-e2cc148b87a8_R/BasicLTI.xml I_a729a1e9-68ab-32db-8e59-7fc049dcc45e_1_R/BasicLTI.xml I_a72cec31-a39f-3a75-93a3-fd631b75564f_R/BasicLTI.xml I_a78118b0-b4fa-3f78-a940-30f86ea36901_R/BasicLTI.xml I_a78afa6f-cd38-3905-918a-f1d8a9316eb8_1_R/BasicLTI.xml I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_1_R/BasicLTI.xml I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_3_R/BasicLTI.xml I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_5_R/BasicLTI.xml I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_7_R/BasicLTI.xml I_a78bc637-6395-38be-9fd9-cc2d8ae95c61_9_R/BasicLTI.xml I_a79c2efa-e3e5-3869-a434-fe01571d9cbc_1_R/BasicLTI.xml I_a819291f-2cc5-3979-af09-43bd10f28e01_1_R/BasicLTI.xml I_a8542ba7-d0fa-37a5-9259-e5f98de3a9a8_1_R/BasicLTI.xml I_a8606a25-40c7-397a-9450-28494c81958d_R/BasicLTI.xml I_a866540c-922d-30d4-9802-2fe9ac224cd3_R/BasicLTI.xml I_a86ac2d1-21f7-3eb7-8360-f5dfe985546e_1_R/BasicLTI.xml I_a8706989-1746-3a72-96ce-a5a90aef30df_R/BasicLTI.xml I_a8a4caec-df5e-3d29-be6f-59e6c0b9f3f2_R/BasicLTI.xml I_a8ac3b40-b1f2-3f60-9f9f-a5b07e7a7a53_R/BasicLTI.xml I_a8b23020-98b3-362c-8549-deeaeb9c3366_1_R/BasicLTI.xml I_a8c13ef1-d428-3a97-9e12-78114499139f_1_R/BasicLTI.xml I_a8c13ef1-d428-3a97-9e12-78114499139f_3_R/BasicLTI.xml I_a8c13ef1-d428-3a97-9e12-78114499139f_5_R/BasicLTI.xml I_a8c13ef1-d428-3a97-9e12-78114499139f_7_R/BasicLTI.xml I_a8c6f799-25fd-3518-9d7a-47df4287f775_R/BasicLTI.xml I_a8dcfc20-8698-3ece-a045-122bdffa52e2_R/BasicLTI.xml I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_1_R/BasicLTI.xml I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_3_R/BasicLTI.xml I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_5_R/BasicLTI.xml I_a8ff73b8-179c-31eb-9eee-49ee26c08e0b_7_R/BasicLTI.xml I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_1_R/BasicLTI.xml I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_3_R/BasicLTI.xml I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_5_R/BasicLTI.xml I_a9163cf0-9bc2-359f-9b5f-81d67247bf32_7_R/BasicLTI.xml I_a9318a58-f29b-34da-8925-bb89d91d50dc_1_R/BasicLTI.xml I_a9318a58-f29b-34da-8925-bb89d91d50dc_3_R/BasicLTI.xml I_a9318a58-f29b-34da-8925-bb89d91d50dc_5_R/BasicLTI.xml I_a94e74ee-a46f-3607-aabb-1a0ca81b13fd_1_R/BasicLTI.xml I_a9547978-60ec-33e6-8467-49a713cce878_1_R/BasicLTI.xml I_a98ce330-3f83-325e-a2a1-edc2316c4885_R/BasicLTI.xml I_a99f4a35-3daa-362e-a8ec-221272f4596a_1_R/BasicLTI.xml I_a9a2d046-5dd7-3543-868e-1acdb9cd32c0_R/BasicLTI.xml I_a9a4dd14-f4be-3130-a14d-b7beca0ae96b_1_R/BasicLTI.xml I_a9b9d0b1-883e-3413-9c8f-f650e18ae8a2_1_R/BasicLTI.xml I_a9c016cd-c29a-305a-991b-003315f4d854_1_R/BasicLTI.xml I_a9c6e247-1f41-3845-9b65-853f878f9874_1_R/BasicLTI.xml I_a9cbbcc8-4355-32e3-8dcf-538c0b03d45b_1_R/BasicLTI.xml I_a9d0907d-d747-3b20-9840-9a899686f37c_R/BasicLTI.xml I_a9f6ea97-b09e-3496-8f7e-5f33f6837842_1_R/BasicLTI.xml I_aa0c058a-113f-3fa8-a7a9-8b294a14878a_1_R/BasicLTI.xml I_aa158de3-d399-3ed9-8193-33f9f0773cf9_R/BasicLTI.xml I_aa2b3b3d-8a69-35f0-88bd-276b639bbd53_1_R/BasicLTI.xml I_aa713813-58a4-3b3a-94fa-45b0b00ba74c_R/BasicLTI.xml I_aa9f91b0-8df6-363c-a978-a11660a5c7ff_1_R/BasicLTI.xml I_aab98cb4-d481-3f43-ad1b-0e43adb7b92a_1_R/BasicLTI.xml I_aad69f1a-aa94-3815-9acf-f7ab3c4a72aa_1_R/BasicLTI.xml I_aad73846-ac80-3bc0-b02d-e34410003eb6_1_R/BasicLTI.xml I_aad9c45d-fed7-3662-b8f3-4831d8052bdf_R/BasicLTI.xml I_aada8eba-a53c-358d-ba71-99bb0aa5c8d9_1_R/BasicLTI.xml I_ab04a4a9-347e-3d96-81c5-fa22fc6072cf_R/BasicLTI.xml I_ab09265e-6f46-3ff9-8c60-314899f98cc9_1_R/BasicLTI.xml I_ab0dfed7-7dc6-389c-b5d5-187b6142f57a_R/BasicLTI.xml I_ab219e0f-5ab1-3de8-8449-ee383b9568aa_1_R/BasicLTI.xml I_ab29d661-a662-3822-a69b-7b286a06d47e_1_R/BasicLTI.xml I_ab4bb24f-4b68-3075-b500-bc2e501c13e8_1_R/BasicLTI.xml I_ab4bb24f-4b68-3075-b500-bc2e501c13e8_3_R/BasicLTI.xml I_ab4bb24f-4b68-3075-b500-bc2e501c13e8_5_R/BasicLTI.xml I_ab4bb24f-4b68-3075-b500-bc2e501c13e8_7_R/BasicLTI.xml I_ab5a3d9f-ccb3-3e9b-97d9-b4af5a8702db_1_R/BasicLTI.xml I_ab5ced21-1d10-365f-85f4-86795db93502_R/BasicLTI.xml I_ab643ddf-0e9d-3feb-826d-005000eea564_R/BasicLTI.xml I_ab6f8f4c-ed9e-3159-88f2-72123f6a5e14_1_R/BasicLTI.xml I_ab71b642-f57a-35b5-8e58-276c569b4764_1_R/BasicLTI.xml I_ab86fe2f-54b7-33ef-b046-13dbd91bc8d3_R/BasicLTI.xml I_aba1dd4b-7a41-3909-b6c1-73f68f79314b_1_R/BasicLTI.xml I_abe29f3d-6a6a-3dfb-b3c6-5ae52656e484_R/BasicLTI.xml I_abfca739-22bf-3021-a47c-43c40f530d42_R/BasicLTI.xml I_ac0de59c-ae94-318a-8d9f-fd8ed4fb641c_1_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_11_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_1_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_3_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_5_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_7_R/BasicLTI.xml I_ac11a97d-2221-33c0-9139-b6a034fa0c42_9_R/BasicLTI.xml I_ac166dfd-2371-3c51-95f0-7611b03b4623_1_R/BasicLTI.xml I_ac170d06-f67a-3234-9ffc-8b8e7ec9d0ce_R/BasicLTI.xml I_ac4be836-bea9-3544-8654-d73fd4f4a01d_1_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_11_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_13_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_15_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_1_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_3_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_5_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_7_R/BasicLTI.xml I_ac5041ef-83d1-3602-99b0-befd80a5d0f8_9_R/BasicLTI.xml I_ac6203f9-6ee7-39a2-943d-90ed6782c24a_1_R/BasicLTI.xml I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_1_R/BasicLTI.xml I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_3_R/BasicLTI.xml I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_5_R/BasicLTI.xml I_ac65251f-2df7-3e6b-8c35-a3d9f969cb6d_7_R/BasicLTI.xml I_ac748cc7-8886-3deb-99cf-75a1e45dfd4a_1_R/BasicLTI.xml I_ac748cc7-8886-3deb-99cf-75a1e45dfd4a_3_R/BasicLTI.xml I_ac748cc7-8886-3deb-99cf-75a1e45dfd4a_5_R/BasicLTI.xml I_ac748cc7-8886-3deb-99cf-75a1e45dfd4a_7_R/BasicLTI.xml I_ac748cc7-8886-3deb-99cf-75a1e45dfd4a_9_R/BasicLTI.xml I_ac84f6f9-8ea2-39db-aff8-c091a7314017_R/BasicLTI.xml I_ac998d1a-9b4e-3b94-a49a-a20673557bc0_1_R/BasicLTI.xml I_acb76ec0-0d92-3215-b66e-4918bed3a20d_R/BasicLTI.xml I_acc547af-f298-3885-8682-d4a8201b12a6_1_R/BasicLTI.xml I_ace94704-a6de-38b1-9b55-0b08d5fcc7cb_R/BasicLTI.xml I_aced7bdd-3688-33c2-b6f1-aede35d1c70d_1_R/BasicLTI.xml I_acee4262-c1e0-3ac4-a03e-282a3c3f1973_1_R/BasicLTI.xml I_ad0f95dc-19d0-3d2f-8d88-cce51a8e4474_1_R/BasicLTI.xml I_ad201f72-f230-327d-9748-b4a4f37aa4d2_1_R/BasicLTI.xml I_ad203183-b6df-36fe-b477-30465d9bdd2c_1_R/BasicLTI.xml I_ad31365f-6d57-3f79-a3ba-59bdb3b84f9c_1_R/BasicLTI.xml I_ad32d290-5799-3d44-a9a0-6a0ece8ca26b_R/BasicLTI.xml I_ad3e0f65-ebd5-3c53-860c-1bc3092ec43e_1_R/BasicLTI.xml I_ad7be270-b728-3bfd-b6f2-987359d216c6_R/BasicLTI.xml I_ad9b5a30-911c-3449-83ba-c29c8a00cf4f_1_R/BasicLTI.xml I_adbca82b-4c2d-36e6-a968-7edcf3527dd2_1_R/BasicLTI.xml I_adbca82b-4c2d-36e6-a968-7edcf3527dd2_3_R/BasicLTI.xml I_adc9cc24-84cd-3768-9656-7ded27ea8b22_R/BasicLTI.xml I_adf49eaa-902c-30c8-9a25-3cc6f40c0f2e_1_R/BasicLTI.xml I_adf953cf-8f1e-3c74-89c6-8b612b815942_1_R/BasicLTI.xml I_ae0adf39-894e-3f24-bf5a-5bb4705202bb_R/BasicLTI.xml I_ae15afb8-1b2d-3aae-8200-4b5da537b53f_R/BasicLTI.xml I_ae32648b-0754-30eb-930d-e4244c626e63_1_R/BasicLTI.xml I_ae36b1ff-b8ff-3b80-b8e0-7691450e6f5a_1_R/BasicLTI.xml I_ae3c6efa-9aec-3436-8675-7ce8a0adeb27_1_R/BasicLTI.xml I_ae5a7166-05f0-3b61-8e9e-aaa6280713f3_R/BasicLTI.xml I_ae6f9f48-0ef3-364f-a099-657b89168798_R/BasicLTI.xml I_ae74bae6-dd22-3cb9-a307-884cb9c0ae53_R/BasicLTI.xml I_ae930321-b592-31ab-9bd9-67cb7a86c4e9_1_R/BasicLTI.xml I_ae930321-b592-31ab-9bd9-67cb7a86c4e9_3_R/BasicLTI.xml I_ae930321-b592-31ab-9bd9-67cb7a86c4e9_5_R/BasicLTI.xml I_ae953c6d-ec64-3001-9307-c62de59e15e7_R/BasicLTI.xml I_ae987415-92ea-3652-8ed7-cdf1880489c5_1_R/BasicLTI.xml I_ae9f1be9-ee27-3a5a-84d3-248bebe99c8e_1_R/BasicLTI.xml I_ae9f998e-2800-35df-b8dd-925758685b93_1_R/BasicLTI.xml I_aeb3edd9-9ca2-30f4-935a-e29d9c00b657_1_R/BasicLTI.xml I_aeb9c1ff-1002-3e3a-8f89-6a31baf8c28d_R/BasicLTI.xml I_aebbd6c5-eaf9-307a-9a62-5b6a821bfb7e_1_R/BasicLTI.xml I_aecb819d-7cda-39b0-88ae-44d64e88db52_R/BasicLTI.xml I_af25d46f-d375-3501-b8e8-3e9921eb9233_1_R/BasicLTI.xml I_af25d46f-d375-3501-b8e8-3e9921eb9233_3_R/BasicLTI.xml I_af2eb1ef-a1ea-38b1-92c6-e66ab8c145fd_1_R/BasicLTI.xml I_af3541fb-acab-3879-8a56-74932583f91a_1_R/BasicLTI.xml I_af3c8c4c-eb94-3de6-8a50-49e17f5fe14c_1_R/BasicLTI.xml I_af42d39f-f8bc-3f40-bdbc-982cc68a8ca9_1_R/BasicLTI.xml I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_1_R/BasicLTI.xml I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_3_R/BasicLTI.xml I_af7d4ab5-5c52-365a-8267-4ef86d5687c2_5_R/BasicLTI.xml I_af894ce6-2ce2-356a-af3a-04a4f804a204_1_R/BasicLTI.xml I_af998683-b5fb-33ce-9081-e80078e8b50c_1_R/BasicLTI.xml I_af9f254c-9559-3dfc-9ca3-4c14dfbe2bb6_R/BasicLTI.xml I_afa7a4b0-099f-3c1e-96ef-f6731dd7f0d1_1_R/BasicLTI.xml I_afc9092f-2037-3f75-8e3e-5689e26b7c1e_1_R/BasicLTI.xml I_afe75a74-0f3e-3918-ab3c-dad55036840f_R/BasicLTI.xml I_b000b965-5bd7-3a7e-9086-cfffcfacc6ca_1_R/BasicLTI.xml I_b011a70d-cd19-3e53-824d-46754331c2d6_1_R/BasicLTI.xml I_b032c866-25cc-39e1-af0c-ccb435a67c7f_R/BasicLTI.xml I_b03ad8e3-4c69-309f-a960-9926abf605c1_1_R/BasicLTI.xml I_b03ad8e3-4c69-309f-a960-9926abf605c1_3_R/BasicLTI.xml I_b06a7222-4342-33b5-b1a3-1f470ded57f6_1_R/BasicLTI.xml I_b0f86d3b-5845-3be3-8420-515d724be521_1_R/BasicLTI.xml I_b0f86d3b-5845-3be3-8420-515d724be521_3_R/BasicLTI.xml I_b0f86d3b-5845-3be3-8420-515d724be521_5_R/BasicLTI.xml I_b0f86d3b-5845-3be3-8420-515d724be521_7_R/BasicLTI.xml I_b0f86d3b-5845-3be3-8420-515d724be521_9_R/BasicLTI.xml I_b101853f-73bf-36f5-95b0-3fe25fab0677_1_R/BasicLTI.xml I_b138ba85-f9ac-377b-8cb6-78f4f4aa92d0_1_R/BasicLTI.xml I_b138ba85-f9ac-377b-8cb6-78f4f4aa92d0_3_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_11_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_13_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_1_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_3_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_5_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_7_R/BasicLTI.xml I_b143e2fa-b2e2-3e1b-8b76-5ea9e0d78130_9_R/BasicLTI.xml I_b174420a-db69-3b25-97f7-ca201dcf1671_1_R/BasicLTI.xml I_b1881962-7c75-37cb-85b0-8fb3173a0817_1_R/BasicLTI.xml I_b1881962-7c75-37cb-85b0-8fb3173a0817_3_R/BasicLTI.xml I_b1881962-7c75-37cb-85b0-8fb3173a0817_5_R/BasicLTI.xml I_b1ad2f07-6884-30ca-ae6d-f01c80c4d13c_1_R/BasicLTI.xml I_b1d6a298-8730-3007-9829-82c167a37bc0_1_R/BasicLTI.xml I_b20b2135-5c09-3975-b176-ea2d0f83fa70_R/BasicLTI.xml I_b21785de-4ea7-392d-82d5-75f068fdd908_1_R/BasicLTI.xml I_b22643fa-9295-307a-ac11-9dace8f9e5de_1_R/BasicLTI.xml I_b2484445-ee68-343e-8a77-cd3e5d98a789_1_R/BasicLTI.xml I_b254feb0-e41d-3227-b475-8c063a235623_1_R/BasicLTI.xml I_b2713402-38c4-3c74-8073-4ed18b4ef7c8_R/BasicLTI.xml I_b2b70507-55bc-3f86-a7aa-ee79fdc14530_1_R/BasicLTI.xml I_b2e518a5-02ac-3314-86ff-43ed0ecc2423_1_R/BasicLTI.xml I_b2f5ab12-d932-3f4f-9c05-731abb5c4fa3_1_R/BasicLTI.xml I_b3034170-b950-3a5d-a5b7-69c5ab33f5bb_1_R/BasicLTI.xml I_b3189f09-1013-37c9-8730-44d64a50cc3f_1_R/BasicLTI.xml I_b329c2cb-33e8-3a9d-9340-f26cbf8586b2_1_R/BasicLTI.xml I_b351329a-cc38-3963-8a51-4f497fe39d11_1_R/BasicLTI.xml I_b3676bcc-3c5e-3bf7-95ed-690ce56ca0ad_1_R/BasicLTI.xml I_b3745911-e4a3-3117-8e7b-baa5f115670d_1_R/BasicLTI.xml I_b380345b-2481-3dd3-beb0-8e7ae704842a_1_R/BasicLTI.xml I_b38f3961-58bb-3ac1-bff0-d46f668e506a_R/BasicLTI.xml I_b393130d-7edb-3438-b8db-7e5c32d976cc_R/BasicLTI.xml I_b399825a-1e92-3c3c-b18e-00d8bdd03737_R/BasicLTI.xml I_b3a6f120-a5ee-38ee-b826-fb92a5c4447e_1_R/BasicLTI.xml I_b3a7cf01-e4ac-346f-9268-e2553e2ee883_R/BasicLTI.xml I_b3b5a778-83f0-3a37-8407-90d57328aa51_1_R/BasicLTI.xml I_b3b8860c-711e-3511-b0f2-d8ec6e724d6a_1_R/BasicLTI.xml I_b3b8860c-711e-3511-b0f2-d8ec6e724d6a_3_R/BasicLTI.xml I_b3b8860c-711e-3511-b0f2-d8ec6e724d6a_5_R/BasicLTI.xml I_b3b8860c-711e-3511-b0f2-d8ec6e724d6a_7_R/BasicLTI.xml I_b3b8860c-711e-3511-b0f2-d8ec6e724d6a_9_R/BasicLTI.xml I_b3dd761c-729e-354f-9aa1-023c2a6e63c4_R/BasicLTI.xml I_b41c285d-b109-3e10-9a5d-2c164da28318_1_R/BasicLTI.xml I_b42829c2-5f31-3d49-bd59-6da72a341f05_1_R/BasicLTI.xml I_b431e36a-6b56-3f06-84ea-4111f27c494b_R/BasicLTI.xml I_b43504e0-9b2f-303f-9d09-82b470926c34_1_R/BasicLTI.xml I_b445a7e7-fb8d-39d0-9f01-b6674e31da4c_1_R/BasicLTI.xml I_b445a7e7-fb8d-39d0-9f01-b6674e31da4c_3_R/BasicLTI.xml I_b449b601-49ef-3fd4-ad1c-fdf223efd291_1_R/BasicLTI.xml I_b45495e7-f31b-3db8-9d09-94dc749a15a7_R/BasicLTI.xml I_b47586d4-bf5f-35e1-92d7-220e0e5590e3_1_R/BasicLTI.xml I_b47586d4-bf5f-35e1-92d7-220e0e5590e3_3_R/BasicLTI.xml I_b4a170cf-dc4c-3dfb-8d45-895b69f0df83_R/BasicLTI.xml I_b4a17283-4f2e-383a-a8c5-a6603b135944_1_R/BasicLTI.xml I_b4c7a511-ffca-3713-8a14-3a57bebc0411_1_R/BasicLTI.xml I_b4d63397-bb3d-3748-b204-30ec98411649_R/BasicLTI.xml I_b4ece2d3-628e-3789-9f4c-5299b18913ee_R/BasicLTI.xml I_b4ed15d9-a5db-3dda-989d-328303e4475d_1_R/BasicLTI.xml I_b4f74a7c-bdac-3885-b286-27d438ceb4ef_R/BasicLTI.xml I_b504a1d7-ae7e-36ed-908b-8441a01f48ad_R/BasicLTI.xml I_b526b5a9-b23f-3380-a858-72f4ccafb5c1_1_R/BasicLTI.xml I_b5351406-a1a0-32c1-b48e-04b386af8099_1_R/BasicLTI.xml I_b56cb907-2c1c-34ef-a6a7-db65f8423ab1_1_R/BasicLTI.xml I_b5847a3b-17de-300c-902c-26f5260e514e_1_R/BasicLTI.xml I_b586b171-21d9-3c16-b0eb-fc8168cda69c_1_R/BasicLTI.xml I_b5895c9f-03a2-399a-8c29-a81697c1199a_R/BasicLTI.xml I_b59861b9-0190-377e-b870-6398672dcc4e_1_R/BasicLTI.xml I_b5b36b19-edc4-373b-9a8a-5525a66046a6_1_R/BasicLTI.xml I_b5ba1ed5-5477-30cf-9e20-2600243f66db_1_R/BasicLTI.xml I_b5cde500-4518-30d7-af20-ea737600655f_1_R/BasicLTI.xml I_b5d17bb1-bbdb-37fb-b0fe-5fbaabed3e6d_1_R/BasicLTI.xml I_b5ff710a-45cf-3409-981b-beea6b92d16e_1_R/BasicLTI.xml I_b6136081-3b1c-3f8a-995d-e6f52a4094ae_R/BasicLTI.xml I_b630566f-bd03-34c2-ac03-09899f86dae4_1_R/BasicLTI.xml I_b630566f-bd03-34c2-ac03-09899f86dae4_3_R/BasicLTI.xml I_b630566f-bd03-34c2-ac03-09899f86dae4_5_R/BasicLTI.xml I_b630566f-bd03-34c2-ac03-09899f86dae4_7_R/BasicLTI.xml I_b630566f-bd03-34c2-ac03-09899f86dae4_9_R/BasicLTI.xml I_b679da6e-0cab-3d4c-a688-043eef2e5a99_1_R/BasicLTI.xml I_b68e9dfc-2957-319b-ab8b-3f0e24bc63a2_1_R/BasicLTI.xml I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_1_R/BasicLTI.xml I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_3_R/BasicLTI.xml I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_5_R/BasicLTI.xml I_b6b02ae9-b3f6-391b-8889-85b1171eaad4_7_R/BasicLTI.xml I_b6b212c2-0e1f-3a81-be64-e40302999b40_R/BasicLTI.xml I_b6cebe15-26d1-313c-b185-1eae6f58f1f1_1_R/BasicLTI.xml I_b6cebe15-26d1-313c-b185-1eae6f58f1f1_3_R/BasicLTI.xml I_b6eb00f9-b982-35d3-a541-421ed231cf3e_1_R/BasicLTI.xml I_b701fc10-d389-319e-b3c5-f32784fa64ab_R/BasicLTI.xml I_b70d6904-0373-3ccf-ab0a-b92c2f33d83a_1_R/BasicLTI.xml I_b714a2d8-0735-3148-bc08-ba0d1f3dd60b_1_R/BasicLTI.xml I_b73abf65-2e86-32d4-91fb-bbb59544618c_1_R/BasicLTI.xml I_b74f277c-0a31-3e3f-8375-a8a2a7350652_1_R/BasicLTI.xml I_b7582907-df1f-3879-931e-a4053ce55186_1_R/BasicLTI.xml I_b76bcd97-038f-33f7-97fc-58d9092ab650_1_R/BasicLTI.xml I_b76bcd97-038f-33f7-97fc-58d9092ab650_3_R/BasicLTI.xml I_b77f1e16-949d-3da6-b142-256c239f6e8e_1_R/BasicLTI.xml I_b782a3a0-97ac-3457-94ef-90d03ebed8a9_1_R/BasicLTI.xml I_b782a3a0-97ac-3457-94ef-90d03ebed8a9_3_R/BasicLTI.xml I_b79ef4b3-4c84-3b5c-8e3a-fa98889086a2_1_R/BasicLTI.xml I_b79f9a12-e3a3-3408-888a-eb6d78a4347c_R/BasicLTI.xml I_b79fa3f4-90fd-32b7-bf88-0a58830d0446_1_R/BasicLTI.xml I_b7a36475-598e-3c78-ba00-745680293772_R/BasicLTI.xml I_b7bec4cb-2482-3412-8cf2-027f0c290a56_1_R/BasicLTI.xml I_b7bec4cb-2482-3412-8cf2-027f0c290a56_3_R/BasicLTI.xml I_b7bec4cb-2482-3412-8cf2-027f0c290a56_5_R/BasicLTI.xml I_b7c5b797-cfe0-3bef-aee3-ed44647c1163_1_R/BasicLTI.xml I_b82b6ee6-9f62-3851-8e1d-38124ee07d0b_1_R/BasicLTI.xml I_b83e4b6c-742c-3c45-8dc8-a6b42d05cb34_R/BasicLTI.xml I_b8430537-331b-3c29-8127-7b939dfac03a_R/BasicLTI.xml I_b8596c0b-82e2-308f-9cd3-3003194cf030_1_R/BasicLTI.xml I_b85dbf82-987b-3a00-b9db-7793b9d30d03_1_R/BasicLTI.xml I_b869bbe1-5d00-3234-9a18-2c1df1515699_1_R/BasicLTI.xml I_b869bbe1-5d00-3234-9a18-2c1df1515699_3_R/BasicLTI.xml I_b8b46124-5625-32a8-b5ab-9c208fd5fa20_1_R/BasicLTI.xml I_b8d1f61d-fcf8-394e-8488-a7d8c26b0f93_1_R/BasicLTI.xml I_b8d1f61d-fcf8-394e-8488-a7d8c26b0f93_3_R/BasicLTI.xml I_b8d1f61d-fcf8-394e-8488-a7d8c26b0f93_5_R/BasicLTI.xml I_b8da3ab4-1e3d-3b44-8f71-f039be064db7_1_R/BasicLTI.xml I_b8f66e09-8ea7-398b-9c65-56aaf60fff65_1_R/BasicLTI.xml I_b8f7ed7f-d8f4-39ef-ad95-6b6bfe6e2913_1_R/BasicLTI.xml I_b8f7ed7f-d8f4-39ef-ad95-6b6bfe6e2913_3_R/BasicLTI.xml I_b8f7ed7f-d8f4-39ef-ad95-6b6bfe6e2913_5_R/BasicLTI.xml I_b8f7ed7f-d8f4-39ef-ad95-6b6bfe6e2913_7_R/BasicLTI.xml I_b8f7ed7f-d8f4-39ef-ad95-6b6bfe6e2913_9_R/BasicLTI.xml I_b91bc427-f21e-3e9e-b222-dd43fe0ce164_1_R/BasicLTI.xml I_b91bc427-f21e-3e9e-b222-dd43fe0ce164_3_R/BasicLTI.xml I_b91bc427-f21e-3e9e-b222-dd43fe0ce164_5_R/BasicLTI.xml I_b91bc427-f21e-3e9e-b222-dd43fe0ce164_7_R/BasicLTI.xml I_b91bc427-f21e-3e9e-b222-dd43fe0ce164_9_R/BasicLTI.xml I_b9266b7a-8106-3a24-9ed8-28e968f9ebc2_1_R/BasicLTI.xml I_b9266b7a-8106-3a24-9ed8-28e968f9ebc2_3_R/BasicLTI.xml I_b9290db7-9ded-39b8-b4ad-7040ef7fee28_1_R/BasicLTI.xml I_b92f3547-b249-3fc3-9f67-f92420aa3af4_1_R/BasicLTI.xml I_b9398f4b-c16f-326b-80eb-5a6faa3866fd_R/BasicLTI.xml I_b94f19de-53a7-3fe8-ae1d-429279591cf4_R/BasicLTI.xml I_b95d228b-586f-3060-b3d0-dca7f75361dc_R/BasicLTI.xml I_b95e1436-b026-34c2-8e07-1152199d9b92_R/BasicLTI.xml I_b96387f2-3230-3429-8d99-eb264bbaf2a4_R/BasicLTI.xml I_b979a642-1442-320d-a3ab-7b5ef44faf3b_R/BasicLTI.xml I_b98dcac5-765e-306e-8fe1-e3a4d2bd2796_R/BasicLTI.xml I_b98efb4d-00df-3a03-b644-911afcefd097_1_R/BasicLTI.xml I_b99d5cda-178a-31e7-8148-73583d114bca_1_R/BasicLTI.xml I_b9b0e843-58ec-35b0-a0e0-4cc1f6469ab0_R/BasicLTI.xml I_b9c7d557-246c-31bc-86a8-eaab9855a1c6_R/BasicLTI.xml I_b9c866c6-10bf-3bff-8291-258c2a6f0ca1_R/BasicLTI.xml I_b9ec5d48-a429-3754-8401-b4aee6cf3df0_R/BasicLTI.xml I_b9ecaa20-a17e-3af2-b41c-eefe3e6c2183_R/BasicLTI.xml I_b9f7efcd-b216-30db-baa3-477e6a3e08fa_1_R/BasicLTI.xml I_b9f904ec-4b0f-35e9-8231-e60b46d532cd_1_R/BasicLTI.xml I_ba025343-5694-3eda-9ee9-48625781bc9a_R/BasicLTI.xml I_ba03178d-c873-3362-afc6-caceed95119c_1_R/BasicLTI.xml I_ba17dfb4-e2ad-3416-9ca6-0d20ec104aec_1_R/BasicLTI.xml I_ba19957a-cd21-37e4-8b0c-ee5d35e54fc0_R/BasicLTI.xml I_ba386059-a994-3f1b-99b8-86d0600878dc_R/BasicLTI.xml I_ba39a36b-6cd1-3e68-83cf-f8461d87ee19_R/BasicLTI.xml I_ba414518-4d29-397e-ac10-771fa508c785_1_R/BasicLTI.xml I_ba685ee7-f3fa-32c9-875c-545da162a443_1_R/BasicLTI.xml I_babf8e65-78f2-37af-a8ff-84884b5fe488_R/BasicLTI.xml I_bac05d80-c0d8-37fc-af96-9482a72c9d90_R/BasicLTI.xml I_bad6e892-4ce8-3e8b-8519-c156af8a1d52_1_R/BasicLTI.xml I_bae9886e-d0ca-3339-a3fa-64eee09d1a6f_R/BasicLTI.xml I_baf2a033-4368-38d1-80a6-89b65e10a63f_R/BasicLTI.xml I_bb06791e-0652-390f-9cae-d66726b8898f_1_R/BasicLTI.xml I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_1_R/BasicLTI.xml I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_3_R/BasicLTI.xml I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_5_R/BasicLTI.xml I_bb095a2f-e2bd-3e7a-9814-5c7761948be6_7_R/BasicLTI.xml I_bb12a4e7-85fe-3aa9-a15e-f77e278b04fc_R/BasicLTI.xml I_bb2a3fcd-b5db-30ee-b611-e8d80330bd4e_1_R/BasicLTI.xml I_bb2c805d-27e4-3aaa-9eeb-7aa2139b3a27_1_R/BasicLTI.xml I_bb552894-349e-3a2a-8d20-376ed062be9d_1_R/BasicLTI.xml I_bb59bc24-f174-32eb-b304-af7b0e0285c6_1_R/BasicLTI.xml I_bb59bc24-f174-32eb-b304-af7b0e0285c6_3_R/BasicLTI.xml I_bb59bc24-f174-32eb-b304-af7b0e0285c6_5_R/BasicLTI.xml I_bb59bc24-f174-32eb-b304-af7b0e0285c6_7_R/BasicLTI.xml I_bb720593-1af6-380c-a0cf-4d7e7630f945_R/BasicLTI.xml I_bb89c206-d548-3a83-9034-be39f0986a40_R/BasicLTI.xml I_bb961e8c-1b91-3e7b-b91e-6e69c91d8eb4_R/BasicLTI.xml I_bbb37ea1-d25b-3fac-8629-794f91571453_R/BasicLTI.xml I_bc0f3b5a-7893-3086-afb6-a31bca58f77e_R/BasicLTI.xml I_bc2196d7-2c73-3666-a264-ef5852c88fda_1_R/BasicLTI.xml I_bc2196d7-2c73-3666-a264-ef5852c88fda_3_R/BasicLTI.xml I_bc2196d7-2c73-3666-a264-ef5852c88fda_5_R/BasicLTI.xml I_bc2196d7-2c73-3666-a264-ef5852c88fda_7_R/BasicLTI.xml I_bc39333c-9a0c-316e-a68c-8b2beba824f0_1_R/BasicLTI.xml I_bc427809-8350-3fde-a971-b68cbd74c4a2_1_R/BasicLTI.xml I_bc4611fb-83bb-315d-b20d-b3d870709c36_1_R/BasicLTI.xml I_bc4f013c-3e79-3472-93dd-cf0c85a04973_R/BasicLTI.xml I_bc63fd3c-18b8-38e1-b17d-e2ab106c06c8_1_R/BasicLTI.xml I_bc70040b-473e-347e-b96d-b7167466632d_1_R/BasicLTI.xml I_bc70040b-473e-347e-b96d-b7167466632d_3_R/BasicLTI.xml I_bc915db9-8d00-39c1-9942-215931d790a4_R/BasicLTI.xml I_bc9d5025-6890-3b7e-a189-2448303af2fb_R/BasicLTI.xml I_bc9df0b5-ad32-36ce-986c-1b61af19e958_R/BasicLTI.xml I_bcaaf8a1-6139-3179-852d-031662d7c2a0_1_R/BasicLTI.xml I_bcaaf8a1-6139-3179-852d-031662d7c2a0_3_R/BasicLTI.xml I_bcaaf8a1-6139-3179-852d-031662d7c2a0_5_R/BasicLTI.xml I_bcaaf8a1-6139-3179-852d-031662d7c2a0_7_R/BasicLTI.xml I_bcaaf8a1-6139-3179-852d-031662d7c2a0_9_R/BasicLTI.xml I_bcd6cec8-98c2-37b0-b96c-1457e2a387d9_1_R/BasicLTI.xml I_bcf4f094-ac6c-3b06-b811-ba43ba562b84_1_R/BasicLTI.xml I_bcf4f094-ac6c-3b06-b811-ba43ba562b84_3_R/BasicLTI.xml I_bcfb248a-f976-37d2-802a-dadf9b6fb5b9_1_R/BasicLTI.xml I_bcfd4a71-de0d-3bdc-9baa-696e3110e41b_1_R/BasicLTI.xml I_bd060b80-6f4f-34c8-a8ec-20e877110434_R/BasicLTI.xml I_bd0871b1-749b-3117-888e-afd3e1de28a3_1_R/BasicLTI.xml I_bd17b405-4280-3573-a1b4-22d1692b55d9_R/BasicLTI.xml I_bd1b3593-02c6-3c9e-a901-cd2cc9d2f16d_R/BasicLTI.xml I_bd1e3852-6e5b-3949-9053-d9ca7c558db3_R/BasicLTI.xml I_bd3f889e-610e-3964-9e9b-96e9f51da0d6_1_R/BasicLTI.xml I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_1_R/BasicLTI.xml I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_3_R/BasicLTI.xml I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_5_R/BasicLTI.xml I_bd451a79-85db-308e-9bdb-d2397eb5d8d3_7_R/BasicLTI.xml I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_1_R/BasicLTI.xml I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_3_R/BasicLTI.xml I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_5_R/BasicLTI.xml I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_7_R/BasicLTI.xml I_bd456b07-c27e-3fff-9a33-d8315fdfe61d_9_R/BasicLTI.xml I_bd50087d-8363-3dbe-8863-829e621df428_1_R/BasicLTI.xml I_bd79416a-f4ec-3a10-b768-59478970de6e_1_R/BasicLTI.xml I_bd97bf78-ee2f-3c94-bf16-e111d8f80315_1_R/BasicLTI.xml I_bddc4059-cb2e-3b6d-af31-f0318e4bfae2_1_R/BasicLTI.xml I_bdec17ed-d784-37f8-a38d-7ff5bc184bf6_1_R/BasicLTI.xml I_be06e44b-cd64-380c-9743-530dd49d0a63_R/BasicLTI.xml I_be0b7d09-43fb-3c5e-934d-1f7fa84ea2b3_1_R/BasicLTI.xml I_be0b7d09-43fb-3c5e-934d-1f7fa84ea2b3_3_R/BasicLTI.xml I_be0b7d09-43fb-3c5e-934d-1f7fa84ea2b3_5_R/BasicLTI.xml I_be0b7d09-43fb-3c5e-934d-1f7fa84ea2b3_7_R/BasicLTI.xml I_be10b73a-4c6f-390f-9e83-4bce3df45d77_1_R/BasicLTI.xml I_be2404b9-8f96-3b52-a9c1-9ae31667a6e4_1_R/BasicLTI.xml I_be2501f1-569e-3a09-bb9c-f7db3ea7918f_1_R/BasicLTI.xml I_be2501f1-569e-3a09-bb9c-f7db3ea7918f_3_R/BasicLTI.xml I_be396531-28c9-3328-a12c-ca0032a02987_R/BasicLTI.xml I_be58a47e-9fa3-3de0-a0f7-6f0acd7afb5b_1_R/BasicLTI.xml I_be75943b-0dc8-3645-91df-580916a6ff1b_1_R/BasicLTI.xml I_be77a2e9-c5c0-3238-9c2d-79ba64e37770_1_R/BasicLTI.xml I_be79f8cc-6ecc-3d54-968e-a8e245b980f2_1_R/BasicLTI.xml I_be7a010b-663f-3a04-8a1e-ec6d6e2572ab_1_R/BasicLTI.xml I_be8721f0-d5c9-34a4-b30f-5c7b6ffe9c2a_1_R/BasicLTI.xml I_be8721f0-d5c9-34a4-b30f-5c7b6ffe9c2a_3_R/BasicLTI.xml I_be8721f0-d5c9-34a4-b30f-5c7b6ffe9c2a_5_R/BasicLTI.xml I_be8721f0-d5c9-34a4-b30f-5c7b6ffe9c2a_7_R/BasicLTI.xml I_be90b3bd-a678-34e4-9ebe-e77032837477_R/BasicLTI.xml I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_1_R/BasicLTI.xml I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_3_R/BasicLTI.xml I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_5_R/BasicLTI.xml I_be933a18-94f6-30e5-9aff-c2afd00b6a6d_7_R/BasicLTI.xml I_be970c3f-80cb-3a00-8931-cd1470059da5_1_R/BasicLTI.xml I_beb22f95-2a2f-3bbc-892f-ab30605cc9f9_R/BasicLTI.xml I_beb45c5a-7d4e-3ff7-ab10-273565394f90_1_R/BasicLTI.xml I_beb515bd-664a-3c43-a9fc-83aa226f71fe_1_R/BasicLTI.xml I_bec1b5c8-b8b3-3d2a-bd3c-1a7b526ff488_R/BasicLTI.xml I_bece1326-cba2-3300-ae6e-89d521343772_1_R/BasicLTI.xml I_bedf96b0-8184-3e3c-865e-818484c9e5f9_1_R/BasicLTI.xml I_bee689ae-c781-3d18-80b6-be4d8edb3222_R/BasicLTI.xml I_bee71be5-7186-3557-b0d4-3ae32d9dc9f2_1_R/BasicLTI.xml I_befde8df-d112-3bfd-90c2-3fcd3b4cac07_1_R/BasicLTI.xml I_befde8df-d112-3bfd-90c2-3fcd3b4cac07_3_R/BasicLTI.xml I_bf417c3d-d03f-355a-88a8-f7a4d316d413_1_R/BasicLTI.xml I_bf4e470c-ff65-30ed-a90b-41502bc4ca85_1_R/BasicLTI.xml I_bf564bc1-d623-32dd-836c-6492874f9cfa_R/BasicLTI.xml I_bf6875f8-89f2-38f8-9add-ca0e4e34a0db_1_R/BasicLTI.xml I_bf6875f8-89f2-38f8-9add-ca0e4e34a0db_3_R/BasicLTI.xml I_bfbfb118-728c-3262-be4c-d969b683da8e_1_R/BasicLTI.xml I_bfbfb118-728c-3262-be4c-d969b683da8e_3_R/BasicLTI.xml I_bfda7666-fd23-39cc-a7be-735326901410_1_R/BasicLTI.xml I_c00ef574-a1ee-3083-b39b-c40921ed9527_R/BasicLTI.xml I_c03cfc48-9aa5-3086-acb1-36cac52e5e03_R/BasicLTI.xml I_c03e2ec3-6dc7-3cff-9a94-ead7881b891d_R/BasicLTI.xml I_c0522548-1855-3d6c-81a8-2b311e33e95e_1_R/BasicLTI.xml I_c0734f8d-1190-3003-8d7d-4efcaa0b05d1_R/BasicLTI.xml I_c0815ea8-7571-3910-9138-408dc4dc2eb1_1_R/BasicLTI.xml I_c0815ea8-7571-3910-9138-408dc4dc2eb1_3_R/BasicLTI.xml I_c087758a-39a9-3d90-8641-21d7e5e0a025_1_R/BasicLTI.xml I_c08b32cc-fdfa-30c7-bdd3-834af9285fa8_1_R/BasicLTI.xml I_c0ae51e5-4e86-35d3-8ad8-4960c8a774b4_1_R/BasicLTI.xml I_c0c4c684-4481-347d-9c91-b5e08839e59a_R/BasicLTI.xml I_c0fd5841-32cc-39bc-914e-29d09e1165e3_1_R/BasicLTI.xml I_c111e4d9-22d6-34d7-b361-bc1cd7dc6ebd_1_R/BasicLTI.xml I_c114dbb5-d8ce-337f-9c70-6449f2331704_1_R/BasicLTI.xml I_c11f845c-4284-3d1c-acab-faad6833ed13_1_R/BasicLTI.xml I_c181fcee-a163-3580-8680-b41e35090817_R/BasicLTI.xml I_c1944749-1b62-3405-84c2-b560ca19d6aa_R/BasicLTI.xml I_c19e170b-2efd-3c9f-933c-0e9988c55f87_R/BasicLTI.xml I_c1a997cb-0857-3bf0-97b6-c5230388def6_1_R/BasicLTI.xml I_c1af4274-afb2-3070-9699-a8afd98e82ab_R/BasicLTI.xml I_c1b23992-fb6c-32af-8373-ef387a625710_R/BasicLTI.xml I_c1d1796e-1f11-3144-86df-f573db2b5eca_1_R/BasicLTI.xml I_c1e7e871-9e1a-338f-ba0c-a7109391a070_1_R/BasicLTI.xml I_c1ed1483-1183-35a8-8161-bf5e46f09824_R/BasicLTI.xml I_c1ef1fc8-6d1d-3b36-9f0e-be267ce46a71_R/BasicLTI.xml I_c1f3d323-617f-36d5-a354-2a07a38e88e4_1_R/BasicLTI.xml I_c1f3d323-617f-36d5-a354-2a07a38e88e4_3_R/BasicLTI.xml I_c1f3d323-617f-36d5-a354-2a07a38e88e4_5_R/BasicLTI.xml I_c1f3d323-617f-36d5-a354-2a07a38e88e4_7_R/BasicLTI.xml I_c1f3d323-617f-36d5-a354-2a07a38e88e4_9_R/BasicLTI.xml I_c1f44852-d28e-34fc-80c3-a83436710176_1_R/BasicLTI.xml I_c1f62c04-0719-36fa-9191-ad7ffd4b2074_1_R/BasicLTI.xml I_c212e9a4-dfeb-34a2-84fb-c105dc7a4112_R/BasicLTI.xml I_c214ade6-e2ba-3d02-9c2a-aca70cbd4d7e_1_R/BasicLTI.xml I_c222b3e2-6654-3028-b756-01f362ace291_1_R/BasicLTI.xml I_c2366512-ed77-3be2-aebb-8b3c280c5819_1_R/BasicLTI.xml I_c2385661-ddcc-353d-bba7-bad7ecda5a96_1_R/BasicLTI.xml I_c2385661-ddcc-353d-bba7-bad7ecda5a96_3_R/BasicLTI.xml I_c2385661-ddcc-353d-bba7-bad7ecda5a96_5_R/BasicLTI.xml I_c2385661-ddcc-353d-bba7-bad7ecda5a96_7_R/BasicLTI.xml I_c2385661-ddcc-353d-bba7-bad7ecda5a96_9_R/BasicLTI.xml I_c23c044a-4cbd-3fbe-963a-f9fb7535d0fb_1_R/BasicLTI.xml I_c23c044a-4cbd-3fbe-963a-f9fb7535d0fb_3_R/BasicLTI.xml I_c23c044a-4cbd-3fbe-963a-f9fb7535d0fb_5_R/BasicLTI.xml I_c24a8e5c-fc92-3964-ae06-c98df0e71a51_R/BasicLTI.xml I_c263659c-26d2-3503-90a6-b99cbf910d34_1_R/BasicLTI.xml I_c263659c-26d2-3503-90a6-b99cbf910d34_3_R/BasicLTI.xml I_c26e9a9d-b2d6-335a-887d-07e829daecbb_1_R/BasicLTI.xml I_c27b18eb-e96d-3e96-a9c4-115987939530_R/BasicLTI.xml I_c27f28b3-6830-38c5-a89a-be4777223696_1_R/BasicLTI.xml I_c27f28b3-6830-38c5-a89a-be4777223696_3_R/BasicLTI.xml I_c287e0aa-54ed-35fb-84cb-dd7a88ec369f_R/BasicLTI.xml I_c28bf417-9c87-3ead-8705-172af50d2a09_1_R/BasicLTI.xml I_c28c9f9e-e2bb-3824-9a0b-b50445044122_1_R/BasicLTI.xml I_c29d7675-ad18-33f5-b096-07322a61ff57_1_R/BasicLTI.xml I_c2ad3e8b-6c8e-3118-9005-97f196482c56_1_R/BasicLTI.xml I_c2bbb4a2-6b16-3e16-97a3-ac5e4c3cf4a9_1_R/BasicLTI.xml I_c2c42752-237a-3604-9401-9b8e2b69a93b_R/BasicLTI.xml I_c2d18b58-92af-3398-b127-f3cd708ea1ca_1_R/BasicLTI.xml I_c2d18b58-92af-3398-b127-f3cd708ea1ca_3_R/BasicLTI.xml I_c2e20cfe-717d-3893-be27-f3e38f1db20d_R/BasicLTI.xml I_c2f1637c-6662-3de5-a13c-368942240256_R/BasicLTI.xml I_c304dd58-d34e-3a16-8d41-1202bcef4922_R/BasicLTI.xml I_c312f6e4-97e5-30ca-bb9e-2c6942aa3108_1_R/BasicLTI.xml I_c31638f4-e47c-3e19-a83e-b4d907624b1c_R/BasicLTI.xml I_c3203549-a4fb-32d6-8ba3-b5397e82655f_1_R/BasicLTI.xml I_c3227d0b-0f80-3f78-b470-6e0a8a8d3d76_1_R/BasicLTI.xml I_c3227d0b-0f80-3f78-b470-6e0a8a8d3d76_3_R/BasicLTI.xml I_c341d828-5929-39da-93be-046eef2d7ee9_1_R/BasicLTI.xml I_c34c61b5-d6f6-39ac-8d30-c6d5a315f56a_1_R/BasicLTI.xml I_c34c61b5-d6f6-39ac-8d30-c6d5a315f56a_3_R/BasicLTI.xml I_c34c61b5-d6f6-39ac-8d30-c6d5a315f56a_5_R/BasicLTI.xml I_c36b22fa-7db7-3082-883a-6e0958c7d6c9_R/BasicLTI.xml I_c36c1be1-e2e2-3b42-92db-bcdcf9395564_1_R/BasicLTI.xml I_c36c1be1-e2e2-3b42-92db-bcdcf9395564_3_R/BasicLTI.xml I_c37b5616-1d7b-3ce0-a1af-8d1071d53328_1_R/BasicLTI.xml I_c3915473-c347-31e1-a2d2-d2e6543075c8_1_R/BasicLTI.xml I_c3ac9f76-9b53-385a-a658-792420f85222_1_R/BasicLTI.xml I_c3bfd019-ce7d-3864-a482-e8752e32f2f8_R/BasicLTI.xml I_c3c2c07d-a505-3e95-9778-0b3b8ac4d24c_R/BasicLTI.xml I_c3c4c705-bea7-3dac-83a2-31238a445400_1_R/BasicLTI.xml I_c3dace06-001e-341b-90b9-6552715767b2_R/BasicLTI.xml I_c3dd9f64-fdef-3510-86ca-0d47305bc781_1_R/BasicLTI.xml I_c4163af4-a5a6-315f-bb35-9808df6851bc_1_R/BasicLTI.xml I_c41afc08-f80d-3687-9cce-68cfc784a9f1_1_R/BasicLTI.xml I_c42a7195-a838-3939-b1ba-ef4a18ff6bdd_1_R/BasicLTI.xml I_c43bf358-40f2-3f36-aa53-a15e9fc38f13_1_R/BasicLTI.xml I_c4435b29-37eb-3dff-aac9-3ea8a4b05faf_R/BasicLTI.xml I_c44b1091-b12a-3930-830d-45686a94c5e5_1_R/BasicLTI.xml I_c450af6b-1726-3ae8-87af-01bf9847d1c1_1_R/BasicLTI.xml I_c464f432-a21a-38c9-9715-0b5fec4d7471_R/BasicLTI.xml I_c46df3cc-2aab-382a-91e2-604e78f64f1d_1_R/BasicLTI.xml I_c489c52f-56b5-3412-a21c-5d67aff4eae4_R/BasicLTI.xml I_c4a6c22d-32fc-34e5-bfe4-f1a5357c90cd_1_R/BasicLTI.xml I_c4b5db3c-e730-3759-ab6d-832adb4728b9_1_R/BasicLTI.xml I_c4b5db3c-e730-3759-ab6d-832adb4728b9_3_R/BasicLTI.xml I_c4d85781-3c97-3224-b912-cc43417497c5_1_R/BasicLTI.xml I_c4e13f99-5a95-3a55-b9a3-e22044b76cc9_1_R/BasicLTI.xml I_c51092e7-3886-3739-860f-8d82045997cd_1_R/BasicLTI.xml I_c51092e7-3886-3739-860f-8d82045997cd_3_R/BasicLTI.xml I_c5121742-2a64-3182-abe4-cb270954fa94_R/BasicLTI.xml I_c52340ac-f4cf-39f8-8b07-d0389eb4c2a2_R/BasicLTI.xml I_c55229d9-f68e-343b-8fe7-92ff5b17ab55_1_R/BasicLTI.xml I_c55668d2-ac3a-3ce2-9b89-d9108710d712_1_R/BasicLTI.xml I_c55be9ed-fdc9-3f9d-aa4e-6d96ca221c08_1_R/BasicLTI.xml I_c577533e-47d5-3d4a-9bfd-b3847508d4d5_1_R/BasicLTI.xml I_c57afe4e-f36c-35d0-b4a5-82d3dd233708_1_R/BasicLTI.xml I_c57afe4e-f36c-35d0-b4a5-82d3dd233708_3_R/BasicLTI.xml I_c57ce2dd-0ba4-3cb2-b2da-c86814c9a586_1_R/BasicLTI.xml I_c58f57a7-cbb8-3f03-9dcf-5cd0e64fe52f_1_R/BasicLTI.xml I_c59b5897-c09c-3c4e-ab57-5c0a480d98e2_R/BasicLTI.xml I_c59def02-852b-3d50-9e35-1e21362eec66_1_R/BasicLTI.xml I_c59def02-852b-3d50-9e35-1e21362eec66_3_R/BasicLTI.xml I_c5a84177-cee0-330f-a98f-4c799de706ec_1_R/BasicLTI.xml I_c5b7acec-5ff3-3448-9f00-9ac018d65f52_1_R/BasicLTI.xml I_c5b7acec-5ff3-3448-9f00-9ac018d65f52_3_R/BasicLTI.xml I_c5c9acd3-9dfd-3ca8-bebf-383c4f59ec36_1_R/BasicLTI.xml I_c5d21efd-e04d-30cb-b86e-b51aa2c52c12_R/BasicLTI.xml I_c5ecc89b-9eb1-3c57-82bd-c0e4c8e0c082_R/BasicLTI.xml I_c5ef2001-e6c4-311d-906c-c25242efb573_1_R/BasicLTI.xml I_c616f239-9043-3459-b973-fb4b703d854e_1_R/BasicLTI.xml I_c619fdb6-058e-3ff3-ab84-1ea7616dd05f_R/BasicLTI.xml I_c63bbe2f-cade-33a9-9822-18d1bbf415e9_R/BasicLTI.xml I_c6418cba-76e5-3920-8ec1-865f2fcd3754_1_R/BasicLTI.xml I_c64cedfd-b924-30fc-9747-0f71ce3f8ce5_1_R/BasicLTI.xml I_c64cedfd-b924-30fc-9747-0f71ce3f8ce5_3_R/BasicLTI.xml I_c64cedfd-b924-30fc-9747-0f71ce3f8ce5_5_R/BasicLTI.xml I_c64cedfd-b924-30fc-9747-0f71ce3f8ce5_7_R/BasicLTI.xml I_c676a175-1c37-30e0-b73c-34ef4bff70f0_1_R/BasicLTI.xml I_c6b980e9-2ad0-3f25-96e9-932936f8bdba_1_R/BasicLTI.xml I_c6c5bafd-0b4e-3d4e-8b07-31b2d5b496ce_1_R/BasicLTI.xml I_c6c5bafd-0b4e-3d4e-8b07-31b2d5b496ce_3_R/BasicLTI.xml I_c6c5bafd-0b4e-3d4e-8b07-31b2d5b496ce_5_R/BasicLTI.xml I_c6c5bafd-0b4e-3d4e-8b07-31b2d5b496ce_7_R/BasicLTI.xml I_c6c5bafd-0b4e-3d4e-8b07-31b2d5b496ce_9_R/BasicLTI.xml I_c6cc635e-0571-3fdc-8eda-18aed177fad2_1_R/BasicLTI.xml I_c6cf26cf-a8a6-3030-bf26-5b7c8f726ceb_1_R/BasicLTI.xml I_c6f38d78-4c47-30d3-9eeb-28d6f357a4fc_R/BasicLTI.xml I_c6f661ac-c788-3448-b276-8ac848df3c93_R/BasicLTI.xml I_c6fb33fa-0a86-33cb-a173-d573104eec4b_1_R/BasicLTI.xml I_c709d471-b869-3cfa-a20f-0c2b3b515e34_1_R/BasicLTI.xml I_c72654cd-2d58-3b2c-8f8f-7dd23b0a54e3_R/BasicLTI.xml I_c72d0416-1425-3eee-b4e4-47247aaf04a3_1_R/BasicLTI.xml I_c736d641-a323-3260-a6e1-6188b510180d_1_R/BasicLTI.xml I_c74b820d-e740-3f0e-a9e7-9e5fa7218c17_1_R/BasicLTI.xml I_c74b820d-e740-3f0e-a9e7-9e5fa7218c17_3_R/BasicLTI.xml I_c7852159-bd83-3ac8-b9be-163094d19c12_1_R/BasicLTI.xml I_c78676ab-982c-3938-976d-47420a9b051d_1_R/BasicLTI.xml I_c78eea3c-287e-36fa-8aa5-c5b7c62ff863_1_R/BasicLTI.xml I_c793f601-151b-3740-ac0a-f613cd48f085_R/BasicLTI.xml I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_1_R/BasicLTI.xml I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_3_R/BasicLTI.xml I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_5_R/BasicLTI.xml I_c7faf00d-fe09-3dd4-9168-d88e3732ff38_7_R/BasicLTI.xml I_c81dcfe7-5f4d-372d-906e-5cd658af14c5_1_R/BasicLTI.xml I_c827b8b6-f77b-3b66-ac75-39dcb621597b_1_R/BasicLTI.xml I_c827b8b6-f77b-3b66-ac75-39dcb621597b_3_R/BasicLTI.xml I_c827b8b6-f77b-3b66-ac75-39dcb621597b_5_R/BasicLTI.xml I_c827b8b6-f77b-3b66-ac75-39dcb621597b_7_R/BasicLTI.xml I_c8333d3d-ced6-3b20-9507-16f16c27f627_1_R/BasicLTI.xml I_c8347719-8ed8-3a53-97cb-2d2b5ff134e8_1_R/BasicLTI.xml I_c8347719-8ed8-3a53-97cb-2d2b5ff134e8_3_R/BasicLTI.xml I_c83b2769-7c6a-368b-8feb-42366031b893_1_R/BasicLTI.xml I_c83b2769-7c6a-368b-8feb-42366031b893_3_R/BasicLTI.xml I_c8476c26-39db-3e3b-ae28-e0d0548f37d1_R/BasicLTI.xml I_c84acd8f-96f8-3140-867b-9549766200bf_1_R/BasicLTI.xml I_c84cd202-1db5-3fed-a747-8caeaeacc3e7_R/BasicLTI.xml I_c8550f83-4def-32de-8d9d-1ee8e3fbda69_1_R/BasicLTI.xml I_c865aaa0-6b5f-38a6-af32-fdffb228c374_1_R/BasicLTI.xml I_c8820ea5-4fd4-3a42-878e-f397aaa606c4_R/BasicLTI.xml I_c89e24fa-e47c-3876-a8cd-d7069abb4764_1_R/BasicLTI.xml I_c8b02ce8-12e5-3070-a55d-2c03360ab268_R/BasicLTI.xml I_c8c66948-0f24-360e-b787-fe85a898d0b8_1_R/BasicLTI.xml I_c920f8ec-65b9-382a-9cd7-a78c754d1827_R/BasicLTI.xml I_c9323849-bd4f-3078-9051-765fdfa4ca82_1_R/BasicLTI.xml I_c94432a2-8834-33d8-a72d-1d8bad6e6bae_1_R/BasicLTI.xml I_c94d0df2-7169-320f-8bad-ba4461e46b62_R/BasicLTI.xml I_c968f220-b8da-3872-a344-3a3d095bf62c_R/BasicLTI.xml I_c9985401-a7f8-374c-a657-86b6b5bdfd02_1_R/BasicLTI.xml I_c9c3e502-91fc-3827-a549-ac8e13ed8b98_1_R/BasicLTI.xml I_c9c7c001-85fe-3e27-a800-44718bdb2fa4_1_R/BasicLTI.xml I_c9cd53d9-5ac6-3041-ab00-6b168b6c7e6a_1_R/BasicLTI.xml I_c9dbb362-59f5-3c85-a9a3-10ef68f19b73_1_R/BasicLTI.xml I_c9fd5166-8297-3c55-af9a-186a885a620b_R/BasicLTI.xml I_ca197d8c-55e8-36ad-bde5-8bfdffa4ebae_1_R/BasicLTI.xml I_ca197d8c-55e8-36ad-bde5-8bfdffa4ebae_3_R/BasicLTI.xml I_ca197d8c-55e8-36ad-bde5-8bfdffa4ebae_5_R/BasicLTI.xml I_ca197d8c-55e8-36ad-bde5-8bfdffa4ebae_7_R/BasicLTI.xml I_ca197d8c-55e8-36ad-bde5-8bfdffa4ebae_9_R/BasicLTI.xml I_ca38e897-4b19-3d57-9c68-891143ed2762_1_R/BasicLTI.xml I_ca4970fa-3abf-3321-8342-7a94cce085cd_1_R/BasicLTI.xml I_ca7060c9-1a75-3279-b11b-cd72f2b27d3a_1_R/BasicLTI.xml I_ca7060c9-1a75-3279-b11b-cd72f2b27d3a_3_R/BasicLTI.xml I_ca897d7d-0a38-3fc9-9cac-d1a35522750f_1_R/BasicLTI.xml I_ca89aa40-4a4f-3c79-98a2-fc3ea39edce6_R/BasicLTI.xml I_ca91f4b1-ac31-32b9-add3-bc615e828ad3_1_R/BasicLTI.xml I_ca91f4b1-ac31-32b9-add3-bc615e828ad3_3_R/BasicLTI.xml I_ca972597-d64f-3c09-b39f-01bd514bf261_1_R/BasicLTI.xml I_cae0a8da-d83f-3f5b-a4c3-3fad7bf16060_R/BasicLTI.xml I_cae2bc94-a8bd-361f-9658-ace4e50d5a72_1_R/BasicLTI.xml I_cae779a5-9f48-30f5-bc04-743e657d94c9_1_R/BasicLTI.xml I_caebf368-399c-3a22-b930-2b64827cebff_1_R/BasicLTI.xml I_caf8e46f-fe85-32ea-a422-cacdc6140eb5_R/BasicLTI.xml I_caf9f790-ee04-32cc-80ad-36782a58097a_R/BasicLTI.xml I_cafdb652-e79f-3c08-9514-9247cdc225ae_1_R/BasicLTI.xml I_cb184758-92f3-36e9-a1f1-2a36cce61549_1_R/BasicLTI.xml I_cb4747b3-5a25-344f-91f0-1470d032632d_1_R/BasicLTI.xml I_cb4c7d7f-6029-349a-aad4-673d7e8c298d_1_R/BasicLTI.xml I_cb54a00f-9f90-3fbd-b5db-1ae72e6c689d_1_R/BasicLTI.xml I_cb57b59d-2195-3483-8dd3-cd915e4fa70e_1_R/BasicLTI.xml I_cb64de54-5586-3030-9d48-5115311b086e_1_R/BasicLTI.xml I_cb8368ed-b839-3345-849e-62a585949e6c_R/BasicLTI.xml I_cb8a1855-a246-39f3-8cd6-90b157829824_1_R/BasicLTI.xml I_cb934a12-8bd0-30de-8c62-d3e285ae15a2_1_R/BasicLTI.xml I_cb9765ac-9a97-3614-bf2f-59382bbd1fc2_1_R/BasicLTI.xml I_cb9c8402-da45-37af-ac13-c02ef2ef5eb2_R/BasicLTI.xml I_cbad5a52-9820-3937-9513-65cdf3e83458_1_R/BasicLTI.xml I_cbb0d454-f1bf-36cc-864f-163d14f4c13c_1_R/BasicLTI.xml I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_1_R/BasicLTI.xml I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_3_R/BasicLTI.xml I_cbe5c6a6-426d-3f8d-b4c5-3547ce49b4a1_5_R/BasicLTI.xml I_cbfe7c61-e044-342e-b4e9-50fe430a661e_R/BasicLTI.xml I_cc24a754-f540-3085-a262-ae48b449dabe_R/BasicLTI.xml I_cc2ae897-f107-3695-99bf-151901d45ebb_1_R/BasicLTI.xml I_cc3133de-c1ae-3231-9a95-de52a811fbbd_R/BasicLTI.xml I_cc43a166-4199-3940-a041-36c47a1fbcb9_1_R/BasicLTI.xml I_cc56061c-ec3f-3eda-b749-a904a113a5a0_1_R/BasicLTI.xml I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_1_R/BasicLTI.xml I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_3_R/BasicLTI.xml I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_5_R/BasicLTI.xml I_cc9b465a-8ddc-3dcd-bd45-d02aa44c3de2_7_R/BasicLTI.xml I_cccf5ef1-ce1b-3d5a-95c5-a28c5b1417ea_R/BasicLTI.xml I_ccdc890c-50d4-3e36-9f9a-5694848cf204_1_R/BasicLTI.xml I_cd04db18-b13e-3184-b848-6bf6e20d6e06_1_R/BasicLTI.xml I_cd1eeece-41de-3209-983b-8d23a8a50b3a_1_R/BasicLTI.xml I_cd503b8e-da9e-3ac6-a1ad-346ad68b5d36_R/BasicLTI.xml I_cd6852b1-0afb-3d58-927d-148cc16ba253_R/BasicLTI.xml I_cd730ced-703d-3d43-b996-bd12d0a63677_1_R/BasicLTI.xml I_cd74281b-3f8c-35dd-80cc-b15ea83aef43_R/BasicLTI.xml I_cd7b9964-8aa7-30b5-8637-14312e803a6d_1_R/BasicLTI.xml I_cdba5d5e-a557-3987-a61b-9d5256bc66d7_R/BasicLTI.xml I_cdc12b3d-605c-34a1-89de-758a54e37cce_R/BasicLTI.xml I_cdc143d6-4f70-3cdd-b1c1-55528e1dc5c0_1_R/BasicLTI.xml I_cdc143d6-4f70-3cdd-b1c1-55528e1dc5c0_3_R/BasicLTI.xml I_cdc5457b-d138-3a80-9a47-1e0005251199_1_R/BasicLTI.xml I_cdca8c06-2e5d-3f15-8b39-46875498febf_1_R/BasicLTI.xml I_cdcbecc1-f403-3efe-bf05-3c4cbcfaf958_1_R/BasicLTI.xml I_cdeaf001-ed8e-3641-b0ab-d2dc50029d08_R/BasicLTI.xml I_cdedba08-c38a-3d61-8564-f2aea9552054_1_R/BasicLTI.xml I_ce076eea-81a8-32f3-a185-a219a2096906_1_R/BasicLTI.xml I_ce0f2c55-87c5-3c34-8b21-ed5ef976fafd_1_R/BasicLTI.xml I_ce1ab9c0-57b5-3111-867d-781eed0607b0_R/BasicLTI.xml I_ce404151-39c9-3da7-ba86-d8e147f3390b_R/BasicLTI.xml I_ce508837-f8eb-3f2f-833a-ac9a3b8b10cf_R/BasicLTI.xml I_ce6f471f-65d4-3bcb-8586-71a3534b063e_R/BasicLTI.xml I_ce7be8f8-9c03-3778-bc9b-ae244e3663af_1_R/BasicLTI.xml I_cea756a0-7477-35e0-9a58-585ff3827895_R/BasicLTI.xml I_ceb8c0fa-756d-31c6-81c1-a33672c88f1c_R/BasicLTI.xml I_cecf61b8-977e-3cfb-852a-fdc64549563a_R/BasicLTI.xml I_ced6225c-5a56-3a42-981a-f90bbdf0fc74_R/BasicLTI.xml I_cedc29db-3883-3c66-a28c-d8ee3b05153a_1_R/BasicLTI.xml I_cf217b64-3ea4-3f65-977d-47a7a0108eea_1_R/BasicLTI.xml I_cf27e7a6-c70c-3c2a-8b02-f89518f347b6_R/BasicLTI.xml I_cf2e1c76-82a1-3b2b-aa8d-71a7358d1458_R/BasicLTI.xml I_cf3e574d-73ca-3287-bb42-766b0a98d079_R/BasicLTI.xml I_cf410100-e336-3fb4-90fa-fa0750894c9a_1_R/BasicLTI.xml I_cf4ead5e-28e4-3f62-8aea-510847eaacaf_R/BasicLTI.xml I_cf5d0431-dda3-31fa-a87f-336429900ef6_1_R/BasicLTI.xml I_cf7947cf-2858-3d10-8b85-27458f3d62b6_1_R/BasicLTI.xml I_cf7c2316-16cc-3869-b5cf-e5177980a880_1_R/BasicLTI.xml I_cf8a07bc-0d84-3d1b-b7f9-474fa12d5a34_1_R/BasicLTI.xml I_cf973166-af81-3eca-989d-2fa6c8c036c9_1_R/BasicLTI.xml I_cfa8d872-af44-3a25-aa13-7445cc5b7590_1_R/BasicLTI.xml I_cfaceb44-e35f-3810-ba05-e1e8eb1705ae_1_R/BasicLTI.xml I_cfc52ff8-c1b9-34b9-85f0-1a327b2e4bbd_1_R/BasicLTI.xml I_cfc52ff8-c1b9-34b9-85f0-1a327b2e4bbd_3_R/BasicLTI.xml I_cfc52ff8-c1b9-34b9-85f0-1a327b2e4bbd_5_R/BasicLTI.xml I_cfc52ff8-c1b9-34b9-85f0-1a327b2e4bbd_7_R/BasicLTI.xml I_cfc52ff8-c1b9-34b9-85f0-1a327b2e4bbd_9_R/BasicLTI.xml I_cfc754da-d8d1-3ea8-b233-0cd9e2cc1451_1_R/BasicLTI.xml I_cfcbfe51-ff78-302c-9f30-fde7b1b3eeaa_1_R/BasicLTI.xml I_cfcbfe51-ff78-302c-9f30-fde7b1b3eeaa_3_R/BasicLTI.xml I_cfcbfe51-ff78-302c-9f30-fde7b1b3eeaa_5_R/BasicLTI.xml I_cfdbb0a7-e6e9-3e30-80f4-781c5737e686_R/BasicLTI.xml I_cfe42e52-2fe0-3f35-acea-9721b09f4246_1_R/BasicLTI.xml I_d00ec9cc-ffb8-381a-838f-ef7f36e0ef67_1_R/BasicLTI.xml I_d0109995-84b7-3194-9073-1946d34af535_R/BasicLTI.xml I_d0187e46-f307-3c08-86e6-9b10ada21909_1_R/BasicLTI.xml I_d0187e46-f307-3c08-86e6-9b10ada21909_3_R/BasicLTI.xml I_d0187e46-f307-3c08-86e6-9b10ada21909_5_R/BasicLTI.xml I_d0187e46-f307-3c08-86e6-9b10ada21909_7_R/BasicLTI.xml I_d028037f-2e0d-336b-8c5e-b4e4eb90b9e0_R/BasicLTI.xml I_d02b01a8-fa92-3111-9fd6-1077c443480c_R/BasicLTI.xml I_d038251f-5ba3-33f9-ba40-fd1efbb67464_1_R/BasicLTI.xml I_d04b13d4-ada0-3afa-9819-6f0856925972_1_R/BasicLTI.xml I_d070031e-3b92-3609-b61f-f5484cfe55be_1_R/BasicLTI.xml I_d075c50a-32b6-3f67-b933-bbc0fb03108c_1_R/BasicLTI.xml I_d0c1d15a-7c81-3263-a087-5b0d44cc8fc9_1_R/BasicLTI.xml I_d0c90471-8ad8-315b-acdb-522db4d460b6_R/BasicLTI.xml I_d0e7513b-4c39-3da9-8b72-b08c882369d2_1_R/BasicLTI.xml I_d0e98a18-6f8a-3d02-a154-cdb59edd6493_1_R/BasicLTI.xml I_d11d1822-4b25-3816-bb40-3006e86a968f_1_R/BasicLTI.xml I_d12a3a10-388f-36fa-8072-3461f7c9f298_1_R/BasicLTI.xml I_d133bf0e-3a69-3579-b06c-173bfce35bc5_1_R/BasicLTI.xml I_d137191a-df10-3706-ac81-017bfc89e829_1_R/BasicLTI.xml I_d14d8f37-2448-36bb-b9bf-4829573307ea_1_R/BasicLTI.xml I_d14d8f37-2448-36bb-b9bf-4829573307ea_3_R/BasicLTI.xml I_d1842033-af23-3ffc-acc8-fe34624197b5_1_R/BasicLTI.xml I_d18ac955-cacd-37f8-9df1-b763c0e22f1c_1_R/BasicLTI.xml I_d1954e02-e5aa-39b5-8ac8-ecc8036f79aa_R/BasicLTI.xml I_d19d8dd9-448e-3975-9ff6-7fef8ebc866b_R/BasicLTI.xml I_d1c67407-ef67-30c4-98b0-0a72e1061e4c_1_R/BasicLTI.xml I_d1c67407-ef67-30c4-98b0-0a72e1061e4c_3_R/BasicLTI.xml I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_1_R/BasicLTI.xml I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_3_R/BasicLTI.xml I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_5_R/BasicLTI.xml I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_7_R/BasicLTI.xml I_d1cf8bc6-ba6a-3226-906a-0fa20084e4ef_9_R/BasicLTI.xml I_d2312865-278f-3341-8a55-96b7808e3400_R/BasicLTI.xml I_d23dcab2-1f77-3d09-9b7b-07c502400554_1_R/BasicLTI.xml I_d23dcab2-1f77-3d09-9b7b-07c502400554_3_R/BasicLTI.xml I_d23f3916-e032-3b6b-a25c-e19786278268_R/BasicLTI.xml I_d250806b-ba4e-37f5-b3e1-bcf3551e2074_1_R/BasicLTI.xml I_d254be79-0462-353c-92cc-822b1171a352_R/BasicLTI.xml I_d2959929-1dd9-356b-9be4-014bbdda6702_1_R/BasicLTI.xml I_d29730dd-8343-3d13-8d61-45132ef03ba7_R/BasicLTI.xml I_d2a81f7e-52c3-3ba9-922b-0169506fc91d_1_R/BasicLTI.xml I_d2be93fd-63eb-3f15-a87e-db0499143573_R/BasicLTI.xml I_d2bf7f3b-c973-3d2a-9d64-0b9c5adb3d05_R/BasicLTI.xml I_d2c441c2-33ff-327c-865d-7e594f091c0d_R/BasicLTI.xml I_d2ff7c8a-7212-30c7-a119-09e9634c24d8_R/BasicLTI.xml I_d30ecf32-da6d-3990-8094-f23e2db5d102_1_R/BasicLTI.xml I_d311e82e-a3c9-31eb-9329-047e62566afa_1_R/BasicLTI.xml I_d31363ef-3457-3d6a-b405-5b76e28ca910_1_R/BasicLTI.xml I_d31363ef-3457-3d6a-b405-5b76e28ca910_3_R/BasicLTI.xml I_d31363ef-3457-3d6a-b405-5b76e28ca910_5_R/BasicLTI.xml I_d32724fd-4c07-3081-83a2-6db1fd476370_1_R/BasicLTI.xml I_d32c1a69-32f9-3ba1-ac8b-df5608d008a3_1_R/BasicLTI.xml I_d32e67f7-15fc-3e33-b855-e9c6910b1938_1_R/BasicLTI.xml I_d33238ce-651c-30eb-8d51-766a17bd2929_1_R/BasicLTI.xml I_d3449044-812c-39ed-ae9f-f0272058987f_R/BasicLTI.xml I_d358f2c4-c860-37e6-abf2-ed5486c43bdc_1_R/BasicLTI.xml I_d35cb8a6-acc5-3082-99d5-0b4aeed261c0_R/BasicLTI.xml I_d36d6486-be77-3f3b-8d91-034eaffcb666_R/BasicLTI.xml I_d37c5546-3b87-3426-9011-15bd69eaa303_R/BasicLTI.xml I_d3823ec5-b730-3734-bcc8-efa932017a8c_1_R/BasicLTI.xml I_d38e48c9-a574-323b-8014-34f7fe8459e2_1_R/BasicLTI.xml I_d39a97f7-9060-33b2-a29b-a12c951a9b30_R/BasicLTI.xml I_d3a49549-6e88-3d83-8248-f60bdeec6d33_R/BasicLTI.xml I_d3b5bcf2-62d3-3ade-8e8f-a3f91106b0ec_R/BasicLTI.xml I_d3d5c419-3863-3fb3-a233-c819afde6dcf_R/BasicLTI.xml I_d3e2b1b8-609e-30fc-a903-aaa59a94fca9_R/BasicLTI.xml I_d3fc11af-5ff9-3675-86b0-bac5f58215b6_R/BasicLTI.xml I_d3fed372-2bf1-3eb7-8f11-521aff5d5bf1_R/BasicLTI.xml I_d407bd81-e5e5-31cf-be0b-120edbf2bdcd_1_R/BasicLTI.xml I_d409c039-d6c7-3239-86e1-e396ce4077dd_1_R/BasicLTI.xml I_d419fef5-444f-3478-a3bb-ce17004d4c2d_R/BasicLTI.xml I_d41c7af1-ab50-3047-a477-89c80ac0c69e_1_R/BasicLTI.xml I_d4273582-d38e-31fd-88a9-3637c637f695_1_R/BasicLTI.xml I_d429fd73-f9b4-3d1d-92a5-951cb3cee16f_R/BasicLTI.xml I_d4414abb-ac99-3c1c-88d3-30efdf504f65_1_R/BasicLTI.xml I_d44f8765-f481-374b-9f33-82970374545b_1_R/BasicLTI.xml I_d45c3495-575d-3898-ab46-3344f59c60cc_1_R/BasicLTI.xml I_d460507c-aafb-3381-aefa-4ee06da17e8e_1_R/BasicLTI.xml I_d462192a-f152-3de4-a311-d34f7331dc42_1_R/BasicLTI.xml I_d469f8d5-7b57-356a-9504-fd34fa2e7426_1_R/BasicLTI.xml I_d488216c-4e4a-39b4-90e0-6824105451bc_1_R/BasicLTI.xml I_d48a2f59-6108-3c3b-b99a-b5356a379a13_1_R/BasicLTI.xml I_d49bb795-7841-31df-9a0f-d7b435c3ae27_R/BasicLTI.xml I_d4b4e967-bd5c-38f0-913c-60d2020e804a_1_R/BasicLTI.xml I_d4b592d6-00e0-3852-a239-c11066e4d286_1_R/BasicLTI.xml I_d4cef1bc-2f54-3955-b318-e87613f326e0_1_R/BasicLTI.xml I_d5073152-9c38-3bbe-b915-bb14cb381848_R/BasicLTI.xml I_d50a221a-c3fc-3442-920b-9c056839b1e5_R/BasicLTI.xml I_d51de676-4ac6-341d-8cab-b586981a9e1e_1_R/BasicLTI.xml I_d526034a-e2e7-34ca-9732-30a060d6a626_1_R/BasicLTI.xml I_d53e0167-62a9-3b2f-93c4-deccc1ff8011_1_R/BasicLTI.xml I_d542e9a7-a4fe-3989-a7c2-2250d3c3593e_1_R/BasicLTI.xml I_d5545555-f3b7-331b-af13-e2295ba26e0a_1_R/BasicLTI.xml I_d5545555-f3b7-331b-af13-e2295ba26e0a_3_R/BasicLTI.xml I_d569dfe4-4c64-3b93-af61-0137495895fd_1_R/BasicLTI.xml I_d569dfe4-4c64-3b93-af61-0137495895fd_3_R/BasicLTI.xml I_d569dfe4-4c64-3b93-af61-0137495895fd_5_R/BasicLTI.xml I_d569dfe4-4c64-3b93-af61-0137495895fd_7_R/BasicLTI.xml I_d56a95d4-adda-3241-b483-db5601241f27_R/BasicLTI.xml I_d58b829d-3515-3f3e-a001-44098130b37c_1_R/BasicLTI.xml I_d58b829d-3515-3f3e-a001-44098130b37c_3_R/BasicLTI.xml I_d59ddd60-f6af-3a93-93ae-19377d3061a0_1_R/BasicLTI.xml I_d5c1191c-6b12-32b3-b3a8-c6a2f9bd61cf_R/BasicLTI.xml I_d5ca39c9-30b7-3060-aa20-f05b48e4407c_R/BasicLTI.xml I_d5e6a48f-026b-3992-85a1-857599370219_R/BasicLTI.xml I_d5e7c07b-88ee-333e-aa3f-1c1601a6cd7e_1_R/BasicLTI.xml I_d5e7f88f-618a-30d2-ba69-04574b60d248_1_R/BasicLTI.xml I_d612805c-cba0-3c51-97fe-2a0f6a9c89b0_1_R/BasicLTI.xml I_d62d4315-7a83-3285-bda8-54e82a94353e_R/BasicLTI.xml I_d6321080-eff7-3afc-9926-f066899ac158_1_R/BasicLTI.xml I_d63379e2-2fc2-3417-9b2e-a901b3014886_1_R/BasicLTI.xml I_d64decf3-bd8e-33d3-b50b-0ecb5766c699_1_R/BasicLTI.xml I_d66928eb-8969-3b11-8028-2a5e031019c9_1_R/BasicLTI.xml I_d66bd8ec-629d-32b4-97a8-df2ab7e8110e_1_R/BasicLTI.xml I_d66bd8ec-629d-32b4-97a8-df2ab7e8110e_3_R/BasicLTI.xml I_d66bd8ec-629d-32b4-97a8-df2ab7e8110e_5_R/BasicLTI.xml I_d66bd8ec-629d-32b4-97a8-df2ab7e8110e_7_R/BasicLTI.xml I_d67d1f2f-7bd0-3c15-ab18-14888b1f5095_1_R/BasicLTI.xml I_d6857b5b-1d4d-33a3-8baf-5c2346b3b11f_R/BasicLTI.xml I_d6880efb-db3a-37c6-8cdc-db08accda90c_1_R/BasicLTI.xml I_d692aeab-840a-3ab1-a014-9bb502d17ba5_R/BasicLTI.xml I_d69bbe15-ccf3-3399-8610-93a8530026ab_1_R/BasicLTI.xml I_d6ac2c11-aad7-36d0-9377-b97026e25295_1_R/BasicLTI.xml I_d6ba4985-2bbd-3240-82bd-99306657b7f9_1_R/BasicLTI.xml I_d6c08fa7-3336-3b3c-9f95-2be055a86c47_1_R/BasicLTI.xml I_d6c3c01c-27e5-3001-a37d-cd478c79c65e_R/BasicLTI.xml I_d6d36d37-061a-33ba-aa9a-05d885e0588d_R/BasicLTI.xml I_d6ee121b-6b40-3dc8-a90f-066445006a88_1_R/BasicLTI.xml I_d700e5b6-e5ca-312a-ac44-b96546c8d64a_1_R/BasicLTI.xml I_d7022258-bb45-320e-8f06-b414f4a9ef6c_1_R/BasicLTI.xml I_d707a88a-d7f1-33db-a34a-9182a8612bfd_1_R/BasicLTI.xml I_d7085e39-f164-3099-a667-11829f5071b6_1_R/BasicLTI.xml I_d71a6794-946b-3910-9126-72c9f6dc9dc3_1_R/BasicLTI.xml I_d75d2a57-79b0-3c23-90ea-179b229d2bb7_1_R/BasicLTI.xml I_d75d2a57-79b0-3c23-90ea-179b229d2bb7_3_R/BasicLTI.xml I_d78279b9-a243-3f94-9e71-062e7335879d_R/BasicLTI.xml I_d7938889-5b85-3f2d-a9cd-9c6e76c20bbb_1_R/BasicLTI.xml I_d7997f21-1a26-3830-a874-46b326101d7f_1_R/BasicLTI.xml I_d79ef592-5ad3-3ca6-8e09-c5ad1f0122d1_R/BasicLTI.xml I_d7a48189-8b2e-33c3-ba02-8ac459fb3b4e_R/BasicLTI.xml I_d7a6a755-f88c-3303-b708-b8b01464a043_1_R/BasicLTI.xml I_d7a7e689-dee7-317a-8f0c-09d180c40d75_R/BasicLTI.xml I_d7ab2408-b8d9-3deb-9c9b-bf8a1ea20d9e_R/BasicLTI.xml I_d7b6032d-ecce-3ca3-b5c8-2caaa454f899_1_R/BasicLTI.xml I_d7d44661-b924-3ca3-8df1-e2e5b91a6d51_R/BasicLTI.xml I_d7ebd1cf-4950-34d6-a3cf-dd6f023e2f61_R/BasicLTI.xml I_d806414c-987e-32f8-8d01-c35e0fab9bf1_R/BasicLTI.xml I_d81bddf5-d3e1-3f15-a269-56f0a778dfb1_1_R/BasicLTI.xml I_d8262970-6a4c-3ba0-b7da-74b59e90df15_1_R/BasicLTI.xml I_d841c546-749d-339e-9130-7dd3f3a32cd7_R/BasicLTI.xml I_d84553ff-7f51-370c-bf2c-36ff18998dba_R/BasicLTI.xml I_d84a2c78-5859-3360-946d-05e9a8d095cb_R/BasicLTI.xml I_d8758308-af74-3e14-8dce-97f0da1e6c4f_1_R/BasicLTI.xml I_d87a3541-874e-37b7-b968-4724ec863010_1_R/BasicLTI.xml I_d88a481f-7439-38a9-a503-10fb2655f859_1_R/BasicLTI.xml I_d89b1dbf-61e4-34ce-b37b-b7a6ad19c463_1_R/BasicLTI.xml I_d89d56e0-b318-38ab-ad8c-1c6f6f9022d6_1_R/BasicLTI.xml I_d8a4e4c8-fd32-3f36-9070-5380abaaefbd_R/BasicLTI.xml I_d8b0ca3a-ab06-3955-9d35-3c5d51dc8748_1_R/BasicLTI.xml I_d8b28a56-b716-35b3-8a37-c626c9bff621_1_R/BasicLTI.xml I_d8bf40e5-c620-3b12-83b6-337f23b9f6a9_1_R/BasicLTI.xml I_d8bf40e5-c620-3b12-83b6-337f23b9f6a9_3_R/BasicLTI.xml I_d8bf40e5-c620-3b12-83b6-337f23b9f6a9_5_R/BasicLTI.xml I_d8bf40e5-c620-3b12-83b6-337f23b9f6a9_7_R/BasicLTI.xml I_d8bf40e5-c620-3b12-83b6-337f23b9f6a9_9_R/BasicLTI.xml I_d8c72a7c-1eb4-350a-bfc0-7ab88a88c85d_R/BasicLTI.xml I_d8caa80d-ea90-3bf4-b0ba-51f2a7196ed8_R/BasicLTI.xml I_d8e0a9b2-b8f7-39ff-9737-ed0b071832c1_1_R/BasicLTI.xml I_d8ed68d7-171f-38c1-bd00-76814cfb7560_R/BasicLTI.xml I_d8f2b205-1abb-3f53-ba22-5b5c26e77440_1_R/BasicLTI.xml I_d8f745d9-0917-3860-9583-19fba0c40dc3_R/BasicLTI.xml I_d8f92912-1b4c-398c-9e37-ca696ed52c1c_1_R/BasicLTI.xml I_d91c56ff-cd7d-3090-a2d5-0fdfc14d530d_R/BasicLTI.xml I_d93c210d-90f0-3bf5-82f1-9cacdfb4b255_1_R/BasicLTI.xml I_d9432b46-95bb-3a56-8e5f-ee443a2c155e_R/BasicLTI.xml I_d94dbefc-c7ef-3ac9-b9c3-7f90e5aeef4c_1_R/BasicLTI.xml I_d96c70ff-4a31-3e3e-9e93-44fe3eab4f40_R/BasicLTI.xml I_d96efbe6-68b3-35ff-a070-b5814bba22f4_1_R/BasicLTI.xml I_d9702554-f42a-385b-90c5-fed5783bdc9f_R/BasicLTI.xml I_d981bc78-05e3-341c-b394-80a7d27c331e_R/BasicLTI.xml I_d98708d3-6bf3-3397-81c9-c92c00ea58a6_1_R/BasicLTI.xml I_d9a9851e-825a-36df-a5be-dafbc80ef7b1_1_R/BasicLTI.xml I_d9a9851e-825a-36df-a5be-dafbc80ef7b1_3_R/BasicLTI.xml I_d9a9851e-825a-36df-a5be-dafbc80ef7b1_5_R/BasicLTI.xml I_d9a9851e-825a-36df-a5be-dafbc80ef7b1_7_R/BasicLTI.xml I_d9a9851e-825a-36df-a5be-dafbc80ef7b1_9_R/BasicLTI.xml I_d9af7d01-8cd2-3b81-bbd5-aa6f1b155c98_1_R/BasicLTI.xml I_d9b62fa2-2ffc-3252-a679-dab93a263335_R/BasicLTI.xml I_d9c5fdf3-25b5-36b4-b4af-73e9dd6e0f17_R/BasicLTI.xml I_d9ef43c0-6965-3a00-bad2-4372fb2826fe_R/BasicLTI.xml I_d9fe85e8-9b13-37a5-86e4-fcd5486e4332_1_R/BasicLTI.xml I_d9fe85e8-9b13-37a5-86e4-fcd5486e4332_3_R/BasicLTI.xml I_d9fe85e8-9b13-37a5-86e4-fcd5486e4332_5_R/BasicLTI.xml I_d9fe85e8-9b13-37a5-86e4-fcd5486e4332_7_R/BasicLTI.xml I_da01b76d-70e8-398f-8c0e-206d308590bd_R/BasicLTI.xml I_da101d4d-2803-36fb-9fdb-999da54a96c4_1_R/BasicLTI.xml I_da1553c0-54e4-3531-956a-fa43121b5f2b_1_R/BasicLTI.xml I_da1904d0-dedd-3968-ab4c-d3a1865ce3e9_R/BasicLTI.xml I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_1_R/BasicLTI.xml I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_3_R/BasicLTI.xml I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_5_R/BasicLTI.xml I_da1d42ea-cc34-3f9f-b91d-01d8572e11e1_7_R/BasicLTI.xml I_da20c386-5833-375d-adf8-26cc16ee68ff_R/BasicLTI.xml I_da2a03a9-eb1c-3d9a-9d33-fa0e108f04a0_R/BasicLTI.xml I_da35b241-026e-371b-92b0-0fe638ffda98_1_R/BasicLTI.xml I_da4339b8-9f84-38cd-84f1-c4d725ea912e_R/BasicLTI.xml I_da47f915-16d7-3ccd-a2e2-13cc4b9bb545_1_R/BasicLTI.xml I_da5c11cf-30aa-3281-b9dd-fee05d86ce62_R/BasicLTI.xml I_da8834ee-542d-30af-9068-babbfd47cc65_R/BasicLTI.xml I_daa9c25e-b8d5-3307-a490-b8e414b16155_R/BasicLTI.xml I_dac500f2-42ed-38f7-a21f-10d308f99e2a_1_R/BasicLTI.xml I_dac5f843-3430-3c80-8bb0-f8794db7c66b_R/BasicLTI.xml I_dacd5594-eab6-331f-94af-fbdb9c8885c6_R/BasicLTI.xml I_dacdc02b-5fb4-3ac2-9645-c59e3cfbeeb6_1_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_11_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_1_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_3_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_5_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_7_R/BasicLTI.xml I_dad84edb-0cf6-3dd7-9791-048cd0fec40f_9_R/BasicLTI.xml I_daf95596-d01f-3dd9-83c5-d7b3061d6c64_1_R/BasicLTI.xml I_db1003f2-0742-3c4f-becb-b5157fe69463_R/BasicLTI.xml I_db2a6d1b-c351-303c-9b80-d77d19d9f76d_1_R/BasicLTI.xml I_db2ab241-e7e3-3b75-bb93-9d4d88bae1f3_R/BasicLTI.xml I_db4df412-117e-36bc-a654-35323b9d372c_1_R/BasicLTI.xml I_db5a6395-ef55-3ff1-ab99-b57cebda9fd7_R/BasicLTI.xml I_db6f6c9f-2797-36d6-b820-a0a0cece462c_R/BasicLTI.xml I_db7b806d-63b3-3421-bca1-2a3e7ec14586_R/BasicLTI.xml I_db82d525-2d20-3624-aa78-6e4e17e51f8b_1_R/BasicLTI.xml I_db893f5b-051c-30e6-b6ed-208e2f8674e4_R/BasicLTI.xml I_db95cab9-7445-336e-8ecc-b4fc33fc7f67_1_R/BasicLTI.xml I_db9640a5-3d2b-3898-83e7-d2cbbe4237f1_R/BasicLTI.xml I_dba50973-103d-3432-9665-b710507a7acd_1_R/BasicLTI.xml I_dba50973-103d-3432-9665-b710507a7acd_3_R/BasicLTI.xml I_dbd543ea-5080-3727-bdf3-54258de2c3cc_1_R/BasicLTI.xml I_dc1c4547-8b56-3450-bb48-4c5bc583278a_1_R/BasicLTI.xml I_dc1c4547-8b56-3450-bb48-4c5bc583278a_3_R/BasicLTI.xml I_dc1c4547-8b56-3450-bb48-4c5bc583278a_5_R/BasicLTI.xml I_dc1c4547-8b56-3450-bb48-4c5bc583278a_7_R/BasicLTI.xml I_dc1c4547-8b56-3450-bb48-4c5bc583278a_9_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_11_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_13_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_1_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_3_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_5_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_7_R/BasicLTI.xml I_dc20d884-e45f-328d-bf4b-862ece0a685e_9_R/BasicLTI.xml I_dc3bb4b8-6f77-3a68-ae41-f652bab563ba_R/BasicLTI.xml I_dc40954a-e21b-37b9-98e2-f61b4032431d_1_R/BasicLTI.xml I_dc45ddb1-029d-39a1-9f43-ad6a8ec41ae1_R/BasicLTI.xml I_dc745182-7a6b-3403-8ecb-bb0f9c15d0a8_1_R/BasicLTI.xml I_dcd3a460-0402-326f-ad6f-7e18775546a2_1_R/BasicLTI.xml I_dcda6558-cd78-358e-92e5-31a144350332_R/BasicLTI.xml I_dce3efd3-ed72-382f-83aa-f6abe3144f93_1_R/BasicLTI.xml I_dce44afe-1fca-3604-b4b7-f0528e3ebe2b_R/BasicLTI.xml I_dd061d25-2229-3756-ba26-3067e276ad19_R/BasicLTI.xml I_dd07db1e-e742-348a-9af0-42abf897b27e_R/BasicLTI.xml I_dd09953b-2438-3022-a68e-c1551975d76c_1_R/BasicLTI.xml I_dd27d8bb-7adf-3f91-ae99-621e476f5523_R/BasicLTI.xml I_dd2cb15a-8c48-369b-9500-737c8d8f8199_1_R/BasicLTI.xml I_dd2cb15a-8c48-369b-9500-737c8d8f8199_3_R/BasicLTI.xml I_dd428243-387f-3f32-b45f-b585a9441a89_R/BasicLTI.xml I_dd4322fe-c54c-3427-8a95-fa4c51494953_1_R/BasicLTI.xml I_dd458eb3-bcca-3c48-8074-ec5bf632dc0a_1_R/BasicLTI.xml I_dd6f8858-3040-322a-8d99-95ee28d353c1_R/BasicLTI.xml I_dd72cee6-0d10-3fdb-98d1-140fd5a46c30_1_R/BasicLTI.xml I_dd798ca4-7893-3c5d-8098-7d4531973251_1_R/BasicLTI.xml I_dd8d8cb4-4ab7-3387-a20f-3fb35202c3f6_R/BasicLTI.xml I_dd8e8fe4-c0e3-3423-bdb2-0745e1ac907b_R/BasicLTI.xml I_dda51557-ffc7-3d06-9580-9d7b1e3dd3a7_1_R/BasicLTI.xml I_ddc8930c-b4a3-3d22-8280-479e464ef7ba_1_R/BasicLTI.xml I_ddc8930c-b4a3-3d22-8280-479e464ef7ba_3_R/BasicLTI.xml I_ddc8930c-b4a3-3d22-8280-479e464ef7ba_5_R/BasicLTI.xml I_ddc8930c-b4a3-3d22-8280-479e464ef7ba_7_R/BasicLTI.xml I_ddc8930c-b4a3-3d22-8280-479e464ef7ba_9_R/BasicLTI.xml I_dddc79e9-0a1c-35f5-8fbb-e26345e3cf45_R/BasicLTI.xml I_dde4e9f1-1d3f-3b00-89e7-7b80c60f1d4b_1_R/BasicLTI.xml I_ddf57590-cfae-345e-89b1-963231f36786_R/BasicLTI.xml I_ddfd3365-04d3-35ae-aec7-8862b325f3a0_1_R/BasicLTI.xml I_de1483d1-40b8-3fc4-9c84-d80e1c4327fa_R/BasicLTI.xml I_de183802-9b91-3998-886f-f8adf2b7d94b_1_R/BasicLTI.xml I_de1d160f-33d3-3411-ac18-92700924fd49_1_R/BasicLTI.xml I_de1d160f-33d3-3411-ac18-92700924fd49_3_R/BasicLTI.xml I_de1d160f-33d3-3411-ac18-92700924fd49_5_R/BasicLTI.xml I_de1d160f-33d3-3411-ac18-92700924fd49_7_R/BasicLTI.xml I_de1d160f-33d3-3411-ac18-92700924fd49_9_R/BasicLTI.xml I_de5129a8-609c-3df0-a29b-e5d141eba7c9_1_R/BasicLTI.xml I_de5492f9-e0d1-3d55-bf9e-786ebbb09737_R/BasicLTI.xml I_de7213f8-19f2-301c-83bf-ce7f76689c2f_1_R/BasicLTI.xml I_de76088d-7712-3ed7-aaa9-6e6136a46c4c_R/BasicLTI.xml I_de771320-5e43-34b3-bbc0-0f8c15838f6e_1_R/BasicLTI.xml I_de8b1c3f-8d25-3fa1-ac50-35c3ca14b825_1_R/BasicLTI.xml I_de8d3393-b8f6-3676-8a3a-ce2980b5e35d_1_R/BasicLTI.xml I_deef28d5-b406-3997-b4fe-6c562ec948d4_R/BasicLTI.xml I_defb04ea-90a4-3fd4-bf8e-0414b6c95a70_1_R/BasicLTI.xml I_df077328-cced-3079-a266-9ec2d2fd4543_R/BasicLTI.xml I_df1e5feb-c57e-30ab-bdc9-0a2d5ef70b5d_R/BasicLTI.xml I_df3e64f2-1257-38aa-9ec4-5c6c7e107ddb_R/BasicLTI.xml I_df4b0835-48f4-330e-a598-64e776ebfee8_R/BasicLTI.xml I_df4fa82e-0646-341e-b964-a4fb0bb767e3_1_R/BasicLTI.xml I_df94a739-c8b3-3a45-848b-0a99acd38756_1_R/BasicLTI.xml I_df990076-5b50-3ec2-b054-a58966fb37ad_1_R/BasicLTI.xml I_df9ae571-67b3-32d9-bf8b-1841dc0f1269_1_R/BasicLTI.xml I_dfab7020-053f-3a27-a05f-68662618fc59_1_R/BasicLTI.xml I_dfab7020-053f-3a27-a05f-68662618fc59_3_R/BasicLTI.xml I_dfd0dcbe-adef-3622-a852-a4c10abc32ef_1_R/BasicLTI.xml I_dfdb70b8-7284-3898-87a2-50e98849db8f_1_R/BasicLTI.xml I_dff54114-2858-3230-bb23-4321ac648335_R/BasicLTI.xml I_e0091c2d-0675-3cb8-a465-bf586891de90_1_R/BasicLTI.xml I_e03e90eb-ecef-37f5-b1c2-202398605bf5_R/BasicLTI.xml I_e03ec9ee-bd38-3d6b-b407-90277719e7a6_1_R/BasicLTI.xml I_e03ec9ee-bd38-3d6b-b407-90277719e7a6_3_R/BasicLTI.xml I_e03ec9ee-bd38-3d6b-b407-90277719e7a6_5_R/BasicLTI.xml I_e03ec9ee-bd38-3d6b-b407-90277719e7a6_7_R/BasicLTI.xml I_e053d002-ad19-369e-954b-aaf9e083049e_R/BasicLTI.xml I_e0553283-54f6-3dce-bcca-9da78b4297f9_1_R/BasicLTI.xml I_e0553283-54f6-3dce-bcca-9da78b4297f9_3_R/BasicLTI.xml I_e06c3602-2093-3235-92f6-f5c3f6afc263_R/BasicLTI.xml I_e06db697-38fe-3f4a-a98b-13b2e3555f46_1_R/BasicLTI.xml I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_1_R/BasicLTI.xml I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_3_R/BasicLTI.xml I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_5_R/BasicLTI.xml I_e084164f-f4fc-3bc8-a0f2-c2f6c5332ca8_7_R/BasicLTI.xml I_e08f0477-f823-301f-81d5-d37995e2915b_1_R/BasicLTI.xml I_e08fb129-7e12-3eb2-8a72-79c0187cea18_R/BasicLTI.xml I_e0abc0e6-7a19-33a0-a337-768d7189c527_R/BasicLTI.xml I_e0b9ca8e-3d00-32e0-b596-828cff8f1623_1_R/BasicLTI.xml I_e0d8df1c-a6f8-3193-ae3f-5f05ffa381b5_1_R/BasicLTI.xml I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_1_R/BasicLTI.xml I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_3_R/BasicLTI.xml I_e0e7f061-c856-3411-92c4-ff4df1b6b34c_5_R/BasicLTI.xml I_e0eddc43-5fe5-3c0a-8b8e-d7942e4b98f0_1_R/BasicLTI.xml I_e1242b9b-cf68-3c5d-a24a-c10cd4f353bc_1_R/BasicLTI.xml I_e13dc471-b1e3-3d7d-857d-947237e898e3_R/BasicLTI.xml I_e1502180-e18b-3dd9-81b3-ac7f71715d84_1_R/BasicLTI.xml I_e17955c7-0edb-346d-b1c9-2a8abfd4bed8_1_R/BasicLTI.xml I_e17955c7-0edb-346d-b1c9-2a8abfd4bed8_3_R/BasicLTI.xml I_e180279d-da8e-35e8-bc94-9ff9e6a30be3_R/BasicLTI.xml I_e186be7c-8db4-3546-a0f7-85eeec2e3cf9_1_R/BasicLTI.xml I_e1b90080-ddb5-30fd-a4a6-de698675526c_1_R/BasicLTI.xml I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_1_R/BasicLTI.xml I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_3_R/BasicLTI.xml I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_5_R/BasicLTI.xml I_e1dfc1c4-9c90-34a2-9b84-2f5b129e180d_7_R/BasicLTI.xml I_e201df7f-1184-3f7e-9e6f-454afbf33afa_1_R/BasicLTI.xml I_e201df7f-1184-3f7e-9e6f-454afbf33afa_3_R/BasicLTI.xml I_e219dc57-f72a-34dc-8e87-40b0fb799256_R/BasicLTI.xml I_e2475064-73a5-31e9-85c3-5a497c71730c_R/BasicLTI.xml I_e248f934-3d8a-3aba-a1f8-359ccfb5a46c_R/BasicLTI.xml I_e24a2937-5780-3d7e-8127-94fe1fd5731e_1_R/BasicLTI.xml I_e25223f0-7688-3068-8da3-f36f67ac99ce_1_R/BasicLTI.xml I_e257015a-99f7-34df-8333-d2cecbd52ba5_1_R/BasicLTI.xml I_e2572884-9277-35a9-becd-cc02c87ebd01_1_R/BasicLTI.xml I_e25b208d-ba3a-34c8-8d5a-a4e0a1410cb1_1_R/BasicLTI.xml I_e270a15c-741f-3a8c-95b0-3910188bad21_R/BasicLTI.xml I_e280870b-e97f-3b80-92e3-22be001b4828_1_R/BasicLTI.xml I_e29c6743-c44d-3168-be9f-6f9138decbff_1_R/BasicLTI.xml I_e2b1c5bd-53f2-322d-b207-372d5bf2ff6b_1_R/BasicLTI.xml I_e2ba1e99-2e50-3454-8124-dc79a8439922_R/BasicLTI.xml I_e2dad8a5-426f-3b4f-96ff-1ea05cbcd098_R/BasicLTI.xml I_e2fd6419-c77d-3520-8fff-7ad8a56e0ba8_R/BasicLTI.xml I_e3023f0e-c07a-3b6c-b82b-bba0f899b04a_1_R/BasicLTI.xml I_e3145127-3fcd-32b1-89f1-03282086f47a_R/BasicLTI.xml I_e326d4bc-a984-3eb1-ae3a-46112aaffeb3_1_R/BasicLTI.xml I_e3456a60-d638-3674-8035-6d7568ca0dc8_R/BasicLTI.xml I_e350a7d4-b124-3768-a182-61354943bfe8_1_R/BasicLTI.xml I_e3646dcd-3241-3c8c-878d-ab2402b26242_1_R/BasicLTI.xml I_e3646dcd-3241-3c8c-878d-ab2402b26242_3_R/BasicLTI.xml I_e372ef18-8c80-3d55-8724-84b012a3147b_1_R/BasicLTI.xml I_e38cd30a-6eb5-3717-bdf4-7b8566edaa7a_R/BasicLTI.xml I_e3966917-a353-3bb7-914c-c18f18630b85_R/BasicLTI.xml I_e3a873e9-6d6d-3d88-96f7-d8ab9058a374_1_R/BasicLTI.xml I_e3a873e9-6d6d-3d88-96f7-d8ab9058a374_3_R/BasicLTI.xml I_e3a873e9-6d6d-3d88-96f7-d8ab9058a374_5_R/BasicLTI.xml I_e3a873e9-6d6d-3d88-96f7-d8ab9058a374_7_R/BasicLTI.xml I_e3b1be1b-a041-3c92-ae56-0beffa46ab80_1_R/BasicLTI.xml I_e3b36f4c-219a-377b-b2b0-07a04655d5df_R/BasicLTI.xml I_e3c30b84-c749-387b-af00-5592ea588973_1_R/BasicLTI.xml I_e3c49d79-2d88-3856-a575-c47453d28962_R/BasicLTI.xml I_e3c8eab0-7d49-3039-b447-0a58b1cb6b85_1_R/BasicLTI.xml I_e3d4ca9d-520d-3bd7-8549-0f56a8cb1241_1_R/BasicLTI.xml I_e3d4ca9d-520d-3bd7-8549-0f56a8cb1241_3_R/BasicLTI.xml I_e3d4ca9d-520d-3bd7-8549-0f56a8cb1241_5_R/BasicLTI.xml I_e3ff89dc-6d21-3ad8-92a5-2f4fcd137136_1_R/BasicLTI.xml I_e4042d58-b16c-3368-a093-5166e9909e15_R/BasicLTI.xml I_e40c6e37-7cd1-3b18-abfb-3ce2b236e5df_R/BasicLTI.xml I_e42ec2ae-31b3-3488-9a00-05811200c0c7_1_R/BasicLTI.xml I_e43a3139-2901-3d38-adff-a01f912cec5b_1_R/BasicLTI.xml I_e43c8f8f-66dc-3839-8205-bd559bc289cc_1_R/BasicLTI.xml I_e43d15e0-22b4-30c0-89cf-2dbc7823577c_1_R/BasicLTI.xml I_e43d15e0-22b4-30c0-89cf-2dbc7823577c_3_R/BasicLTI.xml I_e454b048-c0e2-3c59-b1d9-a50c33c310c9_1_R/BasicLTI.xml I_e454b048-c0e2-3c59-b1d9-a50c33c310c9_3_R/BasicLTI.xml I_e454b048-c0e2-3c59-b1d9-a50c33c310c9_5_R/BasicLTI.xml I_e454b048-c0e2-3c59-b1d9-a50c33c310c9_7_R/BasicLTI.xml I_e458625a-023a-34da-8649-d814c907b255_1_R/BasicLTI.xml I_e45d995d-90cc-3bed-85ee-89ba2cd62a59_R/BasicLTI.xml I_e4640879-38e7-3249-abee-38f3d1135dbf_1_R/BasicLTI.xml I_e472b381-9e0c-3373-92b1-78285f50c662_1_R/BasicLTI.xml I_e47f1a98-3f52-35dc-92e1-8eec9d40e569_R/BasicLTI.xml I_e4939d20-8040-38f5-a6f5-8921b5ef1d32_R/BasicLTI.xml I_e4b6feae-e865-375e-b3a4-b4ec9140daac_1_R/BasicLTI.xml I_e4cc091f-c06d-3751-b45e-b2ad3d628117_R/BasicLTI.xml I_e4cdc1d7-62bf-347a-81e8-e87f72d3043b_1_R/BasicLTI.xml I_e5247cf5-a432-37a1-8a74-797e26e61b83_1_R/BasicLTI.xml I_e5426881-4c71-3687-86d9-df50f38a4e4b_1_R/BasicLTI.xml I_e55b7cd3-02fb-364c-8b8a-409cffdef619_1_R/BasicLTI.xml I_e567bd07-41d4-3194-9311-4ac7b2f9a0f5_1_R/BasicLTI.xml I_e567bd07-41d4-3194-9311-4ac7b2f9a0f5_3_R/BasicLTI.xml I_e567bd07-41d4-3194-9311-4ac7b2f9a0f5_5_R/BasicLTI.xml I_e567bd07-41d4-3194-9311-4ac7b2f9a0f5_7_R/BasicLTI.xml I_e567bd07-41d4-3194-9311-4ac7b2f9a0f5_9_R/BasicLTI.xml I_e57432c1-01e1-33f4-97ec-5c485daf47b1_1_R/BasicLTI.xml I_e5884ed1-bbaf-3134-8c44-118e19823103_1_R/BasicLTI.xml I_e5884ed1-bbaf-3134-8c44-118e19823103_3_R/BasicLTI.xml I_e5884ed1-bbaf-3134-8c44-118e19823103_5_R/BasicLTI.xml I_e58c3df0-6af6-36f3-9274-59d14ae05169_R/BasicLTI.xml I_e58cc8b1-4893-3616-8eba-818e97553273_R/BasicLTI.xml I_e58cccdd-57b7-3c41-95eb-27f90c3505e9_1_R/BasicLTI.xml I_e5b14f61-40e5-3b2e-b5d2-506b9dab80d1_1_R/BasicLTI.xml I_e5b14f61-40e5-3b2e-b5d2-506b9dab80d1_3_R/BasicLTI.xml I_e5b14f61-40e5-3b2e-b5d2-506b9dab80d1_5_R/BasicLTI.xml I_e5b14f61-40e5-3b2e-b5d2-506b9dab80d1_7_R/BasicLTI.xml I_e5b9f85e-8004-34e3-856d-0cc8db72271d_R/BasicLTI.xml I_e621c69f-5da9-3506-81af-407e7ad5096c_1_R/BasicLTI.xml I_e63a5e2c-d272-3b1e-9e9f-3770d5f69d48_1_R/BasicLTI.xml I_e63c6818-27b5-3cf2-a8bf-289bae692023_1_R/BasicLTI.xml I_e641a5b1-298d-3068-9272-e83680f0bd46_1_R/BasicLTI.xml I_e67a2d24-cdec-3d6c-aa3e-8832148c6899_1_R/BasicLTI.xml I_e67a2d24-cdec-3d6c-aa3e-8832148c6899_3_R/BasicLTI.xml I_e6ba0798-99ba-3f43-ba52-c9262595b1c9_1_R/BasicLTI.xml I_e6d64564-cfd3-3fc3-8de1-fffb68ef7d68_1_R/BasicLTI.xml I_e6ed90b9-c771-39ab-895b-a94008a830c3_1_R/BasicLTI.xml I_e6f00709-8405-3b20-8f62-2e8f8f9901b5_1_R/BasicLTI.xml I_e6f00709-8405-3b20-8f62-2e8f8f9901b5_3_R/BasicLTI.xml I_e6f00709-8405-3b20-8f62-2e8f8f9901b5_5_R/BasicLTI.xml I_e6f00709-8405-3b20-8f62-2e8f8f9901b5_7_R/BasicLTI.xml I_e6f00709-8405-3b20-8f62-2e8f8f9901b5_9_R/BasicLTI.xml I_e70495ce-c596-3b13-8543-8c97ccc69108_1_R/BasicLTI.xml I_e724c387-f73e-31af-8200-f84727189def_1_R/BasicLTI.xml I_e72847ef-617f-3cf0-9b7d-f118442781d3_1_R/BasicLTI.xml I_e735824d-abf8-3ac1-b394-38282bbfce4c_1_R/BasicLTI.xml I_e75813fe-b3f4-35e8-93b8-a5f22569800c_R/BasicLTI.xml I_e75fd59b-8a7a-3468-82f4-235811079ece_1_R/BasicLTI.xml I_e7626f4d-13d4-395c-9b9a-ff07615fee9e_1_R/BasicLTI.xml I_e76ea316-3fbf-3825-8640-93cb23d84b6f_R/BasicLTI.xml I_e7a43759-22e2-3a18-b1cd-b51c567a108e_1_R/BasicLTI.xml I_e7aba6f1-f376-3087-ad83-ff3e71a4a837_R/BasicLTI.xml I_e7af946d-8cbb-38e4-8f27-12f1fbaf61fe_1_R/BasicLTI.xml I_e7bb87d0-bee3-3029-9dd3-eec64c3f18e4_R/BasicLTI.xml I_e7d7152b-cddf-3a75-b7b4-b4d9d56df379_1_R/BasicLTI.xml I_e7dc2624-40f9-344d-92e9-7be3c412a301_1_R/BasicLTI.xml I_e7ea33eb-9f2a-3ec3-9e09-42816bc126fa_1_R/BasicLTI.xml I_e7f6124a-f3ae-3e01-b980-916769c3608c_1_R/BasicLTI.xml I_e7f9aa07-39e2-3d67-b7b3-7fb7d8ebac8e_1_R/BasicLTI.xml I_e8102931-055d-3d3a-8c3f-4b973b942092_1_R/BasicLTI.xml I_e82d2157-d280-383e-8f1b-cb52e038d948_1_R/BasicLTI.xml I_e85a121c-c263-3210-b559-889226700b9b_R/BasicLTI.xml I_e8652fca-7c56-3ca0-9971-c047166d9593_1_R/BasicLTI.xml I_e8652fca-7c56-3ca0-9971-c047166d9593_3_R/BasicLTI.xml I_e86ae2a5-1e5d-38ae-b223-30b89e9e7594_1_R/BasicLTI.xml I_e8709bc8-ddad-3a83-ba4b-5611776a0ff5_1_R/BasicLTI.xml I_e8740b37-c72b-3698-abd5-8a93331e589d_R/BasicLTI.xml I_e88cc05e-00c7-3079-97fc-6b701260b58f_R/BasicLTI.xml I_e8bc42b2-f401-33a8-b0b0-a48cac0f64c4_1_R/BasicLTI.xml I_e8c7ded2-43d7-351c-9cf2-e51a99bd1338_1_R/BasicLTI.xml I_e8e34709-9c24-332d-99f5-b87416a96303_1_R/BasicLTI.xml I_e8eba7da-f24b-3830-aab9-c55b603c2cd6_R/BasicLTI.xml I_e8f2f6ab-33ff-3d6e-85a9-182fbf8d7fa6_R/BasicLTI.xml I_e8fd9f73-e004-3b52-9f1a-a344ec53a82a_1_R/BasicLTI.xml I_e8fd9f73-e004-3b52-9f1a-a344ec53a82a_3_R/BasicLTI.xml I_e8fdd133-8a31-37b1-883d-1c64d086888f_R/BasicLTI.xml I_e90fd4e5-c885-3176-9732-de53d585fc4c_R/BasicLTI.xml I_e91b178c-0454-31d0-88c8-a1acee7aa011_1_R/BasicLTI.xml I_e91b73fa-1e82-3f1c-81d9-cf760242185c_R/BasicLTI.xml I_e947b623-4633-331c-8705-3e5bdd7e88fb_1_R/BasicLTI.xml I_e94c1706-e669-3976-ac88-e346e1cbd456_1_R/BasicLTI.xml I_e95f2ae3-a963-3983-b87c-070c79bcec7d_1_R/BasicLTI.xml I_e96423a4-94aa-33a7-b162-1dfcfc72b714_1_R/BasicLTI.xml I_e96423a4-94aa-33a7-b162-1dfcfc72b714_3_R/BasicLTI.xml I_e96423a4-94aa-33a7-b162-1dfcfc72b714_5_R/BasicLTI.xml I_e972027f-2721-3f83-8a7c-9194b194d7fe_R/BasicLTI.xml I_e9734665-7965-376b-a3fa-07717f7ac99d_1_R/BasicLTI.xml I_e97bae59-b461-321f-a9d8-f6eefca339b5_1_R/BasicLTI.xml I_e99bf129-4dd5-3f01-ae4c-eea6b2cbae26_1_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_11_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_13_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_1_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_3_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_5_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_7_R/BasicLTI.xml I_e99ec3e5-13ee-39f3-992e-3b1ca6e6999d_9_R/BasicLTI.xml I_e9ba432a-0ff6-344f-be32-bf2f46ac4c5b_1_R/BasicLTI.xml I_e9c84096-5800-353f-9383-ce6f35eb1a12_1_R/BasicLTI.xml I_e9cc868a-5668-359e-a301-3c3a9ef57756_1_R/BasicLTI.xml I_e9d0cf86-9529-385a-8c97-58c413eed518_R/BasicLTI.xml I_e9ffa0db-f05c-3a11-9bdc-e51b3c7c387b_1_R/BasicLTI.xml I_ea021c85-48ae-3922-812a-c092f7b66e7d_1_R/BasicLTI.xml I_ea03fe3a-566f-3fd8-8257-b6e435b4aacc_1_R/BasicLTI.xml I_ea03fe3a-566f-3fd8-8257-b6e435b4aacc_3_R/BasicLTI.xml I_ea170861-2e65-3220-8ec5-369d5de015da_R/BasicLTI.xml I_ea208300-9562-3d7d-a5c5-ef7699f47021_1_R/BasicLTI.xml I_ea4fb9a7-d354-3562-a6ec-53d44eeff0e1_1_R/BasicLTI.xml I_ea56d0c1-1e21-3faf-9839-b2657d796ee0_R/BasicLTI.xml I_ea5a81cd-638c-3390-8ae4-cead39177798_1_R/BasicLTI.xml I_ea6c8612-dec2-36e0-acb2-742498e5cdb1_R/BasicLTI.xml I_ea6f11b5-52bf-3daf-8c4f-9aeac9618b8c_1_R/BasicLTI.xml I_ea708f55-9c64-3f73-afdf-e07fa4a3532a_1_R/BasicLTI.xml I_ea73fcc6-c286-3fa1-91ed-5706c77ad403_1_R/BasicLTI.xml I_ea78a762-28c4-38e0-8205-f4fc33b9c44a_R/BasicLTI.xml I_ea857453-581c-359e-aba8-12b17c69d3df_1_R/BasicLTI.xml I_eaab0259-a5ea-3009-8a32-49307ab22ccf_R/BasicLTI.xml I_eaac6c1e-7de6-3cf6-9759-6f648031329e_R/BasicLTI.xml I_eaafd7f3-f20a-3d21-95f1-caceea180a37_1_R/BasicLTI.xml I_eae7961f-3c2f-3894-9225-9b86addc603f_1_R/BasicLTI.xml I_eaecfc8f-888b-39be-a039-05a9d919e2b0_R/BasicLTI.xml I_eb1b36f0-d5ed-3a7c-9d12-b1230527bae5_R/BasicLTI.xml I_eb46db5c-3ad7-3eed-8e76-fc76b534f4d7_1_R/BasicLTI.xml I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_1_R/BasicLTI.xml I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_3_R/BasicLTI.xml I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_5_R/BasicLTI.xml I_eb79d02a-e736-3e86-b226-ddebd8dfbaad_7_R/BasicLTI.xml I_eb7f60d0-c747-3b15-ab49-0f779f90f43d_1_R/BasicLTI.xml I_eb9aac5e-02e5-377c-b30b-a0827fe7f06b_1_R/BasicLTI.xml I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_1_R/BasicLTI.xml I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_3_R/BasicLTI.xml I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_5_R/BasicLTI.xml I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_7_R/BasicLTI.xml I_ebb9d9ca-4f3c-3352-bee6-53212e984b61_9_R/BasicLTI.xml I_ebc4378c-1379-3afc-8640-cdae6dcf4281_1_R/BasicLTI.xml I_ebd04a7e-698c-3341-863f-27b8efa286ba_1_R/BasicLTI.xml I_ebd1f0f6-15b8-34dc-bc3f-a2072e4fff13_1_R/BasicLTI.xml I_ebe78c47-464c-30c2-814b-b7a61daaa071_1_R/BasicLTI.xml I_ebf9957e-c76f-3993-a15d-402d4eeeb78b_R/BasicLTI.xml I_ebfe2146-28c5-3484-8142-3db2af53df16_R/BasicLTI.xml I_ec0d9550-959e-37d8-badd-54efe86cadfa_1_R/BasicLTI.xml I_ec12d1ea-c2b1-334c-bc37-968d3e0b2bf8_1_R/BasicLTI.xml I_ec1b7a55-5e1e-3a2e-84c9-76b52b44567a_R/BasicLTI.xml I_ec256399-5e79-3df1-9c14-2f83aa5aedfa_1_R/BasicLTI.xml I_ec3d95d5-8a88-3aa1-803f-97b15374266a_R/BasicLTI.xml I_ec3e088e-c85c-38a3-95e8-79276d4edc8c_1_R/BasicLTI.xml I_ec48ca91-bbe4-3702-b973-8f47838e54f2_1_R/BasicLTI.xml I_ec492d28-cd06-37dd-9f21-44a31d356f5d_R/BasicLTI.xml I_ec71b106-f207-3bbb-a4ac-cda7eb3d8c6c_1_R/BasicLTI.xml I_ec89d564-1abf-3a5d-8dd0-b28bef457e1e_1_R/BasicLTI.xml I_ec9dcb43-8504-3e24-b7f2-a5c3b1bee409_R/BasicLTI.xml I_ecb48335-029a-318a-98f4-ead1408ea99d_1_R/BasicLTI.xml I_ecb53428-a813-3c4d-934f-db0761077ec5_1_R/BasicLTI.xml I_ecce759f-d406-38e0-8940-e2802c77f2b7_R/BasicLTI.xml I_ecf11259-9336-37d9-b8f3-348b20a88b55_1_R/BasicLTI.xml I_ed0208f4-29fd-3289-a527-ebab7881524b_1_R/BasicLTI.xml I_ed0208f4-29fd-3289-a527-ebab7881524b_3_R/BasicLTI.xml I_ed0680ea-e83e-31cb-b4be-75d4ba4c0612_1_R/BasicLTI.xml I_ed093469-38d3-322d-ac16-026967b27096_1_R/BasicLTI.xml I_ed0fcac1-0b0d-36d3-b8a4-35d3f9b4113f_1_R/BasicLTI.xml I_ed177bd4-7c78-3804-b164-0074634194ec_1_R/BasicLTI.xml I_ed1eaf6d-c755-3f84-83a9-044fec9985f3_R/BasicLTI.xml I_ed2d4317-72b9-3395-9f0e-782d40b0657c_1_R/BasicLTI.xml I_ed2fd960-265b-350a-a152-e60639591a24_1_R/BasicLTI.xml I_ed48bd5c-96fa-3233-8b50-e7bdd85cf84a_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_11_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_13_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_1_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_3_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_5_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_7_R/BasicLTI.xml I_ed4d1477-5ff5-39ea-b8fb-0dd87320eb70_9_R/BasicLTI.xml I_ed524190-f906-38fa-8b99-7d2cfa41db66_1_R/BasicLTI.xml I_ed524190-f906-38fa-8b99-7d2cfa41db66_3_R/BasicLTI.xml I_ed524190-f906-38fa-8b99-7d2cfa41db66_5_R/BasicLTI.xml I_ed615059-8887-3a30-8487-986f30d436b6_R/BasicLTI.xml I_ed73e82c-3a01-3022-90f4-e05acb0c0a9d_R/BasicLTI.xml I_ed8f6314-69b4-3d9f-9834-8eb0f328fdee_R/BasicLTI.xml I_edcdeccb-2198-3210-91c5-10f6437acb29_1_R/BasicLTI.xml I_edce5c63-9301-32ea-aa0a-55e48aa10e0a_1_R/BasicLTI.xml I_edce5c63-9301-32ea-aa0a-55e48aa10e0a_3_R/BasicLTI.xml I_edd464da-7d89-3e80-9553-a59319fa3a01_1_R/BasicLTI.xml I_ee02f4aa-a8c6-33d8-b3d2-2a3fa70bafd5_1_R/BasicLTI.xml I_ee079adc-bddc-3409-88c6-b74f40503ebc_1_R/BasicLTI.xml I_ee079adc-bddc-3409-88c6-b74f40503ebc_3_R/BasicLTI.xml I_ee079adc-bddc-3409-88c6-b74f40503ebc_5_R/BasicLTI.xml I_ee2b67fe-fc26-317c-bc78-e5955d7813aa_1_R/BasicLTI.xml I_ee361b19-4cd5-3167-8d20-497c70645f1c_R/BasicLTI.xml I_ee3cbae3-3377-316e-8290-f051c1c7540d_R/BasicLTI.xml I_ee4de0e5-4b84-3fcf-b363-ed45f576e6ff_R/BasicLTI.xml I_ee50ad10-2121-3392-80f0-41177da02694_R/BasicLTI.xml I_ee514415-4720-3726-ab24-dbb58f15a7c1_1_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_11_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_13_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_1_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_3_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_5_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_7_R/BasicLTI.xml I_ee5450cd-6785-321a-b2a0-1b5265e384a1_9_R/BasicLTI.xml I_ee7a6e71-16b0-31ca-a3d0-9cd68250e8e2_R/BasicLTI.xml I_ee856b61-6392-3767-b9ff-6064a5a2daa5_1_R/BasicLTI.xml I_eea122eb-636c-3d2c-a9d9-a652541da5fd_1_R/BasicLTI.xml I_eea5884b-2dbb-3dd4-b69d-2d662730faa2_R/BasicLTI.xml I_eea7dee8-5619-3c1c-82ab-abb6b4c40641_1_R/BasicLTI.xml I_eea7dee8-5619-3c1c-82ab-abb6b4c40641_3_R/BasicLTI.xml I_eeabf8a0-0d09-316c-ada6-77f368b6e90f_R/BasicLTI.xml I_eeb7561e-d2db-39c0-b555-8992890c484b_R/BasicLTI.xml I_eebc486e-9f3d-34a7-8851-d3588c895c96_R/BasicLTI.xml I_eec58aa7-04ba-3718-997a-7bf68a07aff3_R/BasicLTI.xml I_eed73f95-3524-3af9-8e1c-3dbb330c711a_R/BasicLTI.xml I_eed89784-1d51-3a4a-ac3b-41bc5aa39188_1_R/BasicLTI.xml I_eed8d5ac-4b89-308b-b1ca-2b959266c968_1_R/BasicLTI.xml I_eedde0f2-12c7-3e3e-9bf0-a3399e40c5ea_1_R/BasicLTI.xml I_eeeb6f7d-8ce6-3312-a2e0-774835e4e735_1_R/BasicLTI.xml I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_1_R/BasicLTI.xml I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_3_R/BasicLTI.xml I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_5_R/BasicLTI.xml I_ef14ed8e-4414-3aca-8714-b53ad5f4551f_7_R/BasicLTI.xml I_ef249deb-4a84-3b95-a351-a6a4e4515e26_1_R/BasicLTI.xml I_ef2f0c0e-a6fe-3a06-b768-d775f0218580_R/BasicLTI.xml I_ef504192-471f-38ee-87d0-7ddea47d0b6f_R/BasicLTI.xml I_ef560b84-a939-3a38-8c9d-54048ab02721_R/BasicLTI.xml I_ef7e9864-bb5e-348a-8ef5-21a52ba3a3ee_1_R/BasicLTI.xml I_ef7e9864-bb5e-348a-8ef5-21a52ba3a3ee_3_R/BasicLTI.xml I_ef8a343e-d9f3-395d-a577-6eb672083a34_R/BasicLTI.xml I_ef9217f4-b77f-3209-91d2-e041bdb255a2_1_R/BasicLTI.xml I_efa69277-4fdf-3bc6-9c47-de49b37f6df3_R/BasicLTI.xml I_efb687f0-a573-3994-b8f2-1d164419973d_R/BasicLTI.xml I_efcc3ff0-d301-37bf-b233-cc58a7b2713a_1_R/BasicLTI.xml I_efe9d394-1186-3364-b73f-5c9d49497ea7_R/BasicLTI.xml I_f0088c89-77f5-309f-997b-6e9967f3acae_1_R/BasicLTI.xml I_f02ed784-b8e2-3856-a397-a19328bfc3d8_1_R/BasicLTI.xml I_f0462681-6f52-3421-a3df-134daf75548e_1_R/BasicLTI.xml I_f06c9f67-bad7-354f-a7eb-f2544bca4dc4_R/BasicLTI.xml I_f07247f1-1a51-394d-9948-4cca19fc5716_1_R/BasicLTI.xml I_f0861df0-f052-3d22-9be7-e5066e7bc6ba_1_R/BasicLTI.xml I_f0931746-69df-3ab9-b866-10e476e1d11d_1_R/BasicLTI.xml I_f098adee-6363-377e-8efd-e81bd49d9476_1_R/BasicLTI.xml I_f0abd240-296d-3ea3-a92a-0127695a38f4_1_R/BasicLTI.xml I_f0c83e71-9d23-3403-b35e-e297ebbabe97_1_R/BasicLTI.xml I_f0cdc697-a0e7-34e0-81cc-3740882c5ebc_R/BasicLTI.xml I_f0d928d5-4145-3bd7-9561-91fb184efc28_1_R/BasicLTI.xml I_f0e2d5b6-857e-33aa-8e95-2497caaef957_1_R/BasicLTI.xml I_f0f836c8-f2a0-32ec-8dda-28afebcf8757_1_R/BasicLTI.xml I_f1112681-d705-31f1-8478-f0a9a7a675b7_1_R/BasicLTI.xml I_f1112681-d705-31f1-8478-f0a9a7a675b7_3_R/BasicLTI.xml I_f113d1da-c019-3860-becc-5ff8d01f3943_R/BasicLTI.xml I_f119f2ae-2aca-3060-a994-d3087584f662_R/BasicLTI.xml I_f11b567f-f7e0-3050-8931-1eca014f9f99_1_R/BasicLTI.xml I_f11b567f-f7e0-3050-8931-1eca014f9f99_3_R/BasicLTI.xml I_f137faeb-3704-39cb-9a48-fe8fb95b417a_1_R/BasicLTI.xml I_f1447238-4bcf-30c5-9287-e9b746450378_R/BasicLTI.xml I_f19148f3-ca39-3e7a-9104-fe57ec197eef_R/BasicLTI.xml I_f195acb7-fe2c-3c8d-9d21-9a745f37a3b6_1_R/BasicLTI.xml I_f1a90deb-f29f-3c84-b31f-03563653c796_1_R/BasicLTI.xml I_f1b4e8a8-be30-3d2c-9f01-11f5847bba9e_1_R/BasicLTI.xml I_f1c7927f-0ac2-3033-8d59-84d4fe155831_1_R/BasicLTI.xml I_f1d470ef-e10f-345e-bf8c-1d3451ac007d_R/BasicLTI.xml I_f1ddd6c6-746f-30f6-a4df-d6f82dab9aec_R/BasicLTI.xml I_f1e536f7-b9ab-39eb-b2b7-871cfefb68d6_1_R/BasicLTI.xml I_f1fc016a-fa53-3525-991e-871caa5351a7_1_R/BasicLTI.xml I_f1fc016a-fa53-3525-991e-871caa5351a7_3_R/BasicLTI.xml I_f1fc016a-fa53-3525-991e-871caa5351a7_5_R/BasicLTI.xml I_f1fc016a-fa53-3525-991e-871caa5351a7_7_R/BasicLTI.xml I_f202d816-01c9-36a4-962f-390dadf80301_1_R/BasicLTI.xml I_f20ee696-f5a8-3470-94ea-869195e1577b_1_R/BasicLTI.xml I_f20ee696-f5a8-3470-94ea-869195e1577b_3_R/BasicLTI.xml I_f20ee696-f5a8-3470-94ea-869195e1577b_5_R/BasicLTI.xml I_f20ee696-f5a8-3470-94ea-869195e1577b_7_R/BasicLTI.xml I_f20fcd36-8039-35b2-9346-8ef7afc9f8e7_R/BasicLTI.xml I_f21d9925-d137-3570-a5f0-3b521875af3c_R/BasicLTI.xml I_f21f10a8-1187-3774-b065-ff1922d40105_R/BasicLTI.xml I_f237fcb5-5834-31b4-872e-7872f7016eab_1_R/BasicLTI.xml I_f247529d-a799-31c7-9ba7-1d6e4b8c0611_R/BasicLTI.xml I_f25fa8df-af58-3bee-8301-aee6b40a0e9e_1_R/BasicLTI.xml I_f25fa8df-af58-3bee-8301-aee6b40a0e9e_3_R/BasicLTI.xml I_f2645bc3-3d7e-3ecb-9954-1adb1976cc7e_R/BasicLTI.xml I_f271454a-9b07-33fc-a2ed-fef18184cd56_1_R/BasicLTI.xml I_f27955a1-6efd-3909-a084-94a342d99112_1_R/BasicLTI.xml I_f29aab26-5f7d-33f8-8600-7a0fe47d5425_1_R/BasicLTI.xml I_f29aab26-5f7d-33f8-8600-7a0fe47d5425_3_R/BasicLTI.xml I_f2a0ea37-e3d7-3ae2-b4f0-c45a6dd44137_R/BasicLTI.xml I_f2a1945f-e207-3f54-b76c-4641b2dded98_R/BasicLTI.xml I_f2b412e6-9476-3849-b663-a8acad10747a_1_R/BasicLTI.xml I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_1_R/BasicLTI.xml I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_3_R/BasicLTI.xml I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_5_R/BasicLTI.xml I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_7_R/BasicLTI.xml I_f2d79a74-558e-39b9-9341-62f97ffd5f6d_9_R/BasicLTI.xml I_f2de851e-e13e-3020-9942-7aa86b0547f4_1_R/BasicLTI.xml I_f2e6a232-5372-38e5-96b2-a100fad4ec8b_1_R/BasicLTI.xml I_f2ebf4fb-b325-3f00-8144-3276b526f5b5_1_R/BasicLTI.xml I_f2ec9fd6-dbf9-3459-bdf3-05b3bc82fe93_1_R/BasicLTI.xml I_f2feab97-49c2-3917-82c8-5f94c9a216be_R/BasicLTI.xml I_f30f6a18-017e-3274-aeb4-b75463a7c59d_R/BasicLTI.xml I_f31d2f53-c569-3c58-ae4b-bbbb45d69c22_R/BasicLTI.xml I_f336356b-b33d-38ad-9380-00853e05cc50_R/BasicLTI.xml I_f33e9ade-945e-37fd-ad05-e0adaf95d159_1_R/BasicLTI.xml I_f3454500-6db5-379c-9ab0-a2f7244fb3cf_1_R/BasicLTI.xml I_f3454500-6db5-379c-9ab0-a2f7244fb3cf_3_R/BasicLTI.xml I_f3454500-6db5-379c-9ab0-a2f7244fb3cf_5_R/BasicLTI.xml I_f3454500-6db5-379c-9ab0-a2f7244fb3cf_7_R/BasicLTI.xml I_f3653c06-4de5-3bb0-b8e3-d1455690bc26_1_R/BasicLTI.xml I_f38b1cee-d38b-3a0e-b0e3-9727d9641902_1_R/BasicLTI.xml I_f39633de-2a89-324f-bbf8-6304f3c0e516_R/BasicLTI.xml I_f397adf9-c47a-32ed-9fc8-f3832614797a_1_R/BasicLTI.xml I_f398ed25-e096-36dc-86b1-e47df0c72ffa_1_R/BasicLTI.xml I_f3b181a0-67f7-363d-9b7b-c079f981860b_1_R/BasicLTI.xml I_f3be4cca-d8fa-330b-9cf0-f3a9321bcff4_1_R/BasicLTI.xml I_f3c12cbc-a5ec-30b7-84e8-c9ffcaa84d8b_1_R/BasicLTI.xml I_f3d6752d-33b0-3106-bb0a-629576977892_1_R/BasicLTI.xml I_f3d93d81-9b6e-3f82-9a98-76c03a06d4c7_1_R/BasicLTI.xml I_f3e8b1d7-463a-385c-8798-be0f57c4a516_1_R/BasicLTI.xml I_f3eb771a-2ca8-3ea5-b07f-89a7ac4c56f2_1_R/BasicLTI.xml I_f3f0cbc2-4a80-3619-8bd6-c9e1f61ddf06_R/BasicLTI.xml I_f40a8e1f-4ec3-31f2-a82e-6f3e986ecb42_1_R/BasicLTI.xml I_f40e2674-3f43-3dda-b173-188f30c47c35_R/BasicLTI.xml I_f418d1d9-dc6f-3d72-9c56-d697dad8786d_1_R/BasicLTI.xml I_f41ac58e-bc2a-3af2-8c41-9251e57a15e8_R/BasicLTI.xml I_f41d1547-c01c-345d-b5e5-92b3deb14f30_1_R/BasicLTI.xml I_f4287660-6f1d-3618-992f-d7f91af905c3_1_R/BasicLTI.xml I_f433a129-3f9e-3454-a516-714dda5b3937_R/BasicLTI.xml I_f45c676b-2e86-31d2-903d-da563df836dc_1_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_11_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_13_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_1_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_3_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_5_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_7_R/BasicLTI.xml I_f462e1bc-c55d-3fd1-9f00-fef489e99097_9_R/BasicLTI.xml I_f4702a5f-d45e-38a0-98b6-c4358d40c0bb_1_R/BasicLTI.xml I_f4cbff44-239d-3589-9971-f784273cc576_1_R/BasicLTI.xml I_f4cbff44-239d-3589-9971-f784273cc576_3_R/BasicLTI.xml I_f4cbff44-239d-3589-9971-f784273cc576_5_R/BasicLTI.xml I_f4cbff44-239d-3589-9971-f784273cc576_7_R/BasicLTI.xml I_f4f11822-66b3-387b-94c1-03d761a903d2_1_R/BasicLTI.xml I_f4f34f78-0255-3858-88d3-07a668a1fec6_R/BasicLTI.xml I_f4f8ef84-b7e9-3612-b51a-f87addf9d65b_R/BasicLTI.xml I_f501c4f8-f852-3253-a2b2-ef8a38aff907_R/BasicLTI.xml I_f502e955-b7dc-30cc-939a-d97d8ddae8b5_1_R/BasicLTI.xml I_f502e955-b7dc-30cc-939a-d97d8ddae8b5_3_R/BasicLTI.xml I_f506c8a1-5f0c-35b1-a65e-6254affc554b_R/BasicLTI.xml I_f50c0106-1f8e-3bdd-a8dd-9f5b0a739461_1_R/BasicLTI.xml I_f51dd019-da22-3395-b7ce-260b69d2a259_R/BasicLTI.xml I_f5205e7c-1f06-3ab9-8216-a49125acc8d5_1_R/BasicLTI.xml I_f5277e46-c0cc-3d80-a113-d15c0a4af9f3_R/BasicLTI.xml I_f52f735a-24fd-363c-96c9-7b5249725a35_1_R/BasicLTI.xml I_f530e6a1-1d20-384f-ae60-068987118edf_R/BasicLTI.xml I_f5318bd6-eb38-3236-8680-a36ff3849a83_1_R/BasicLTI.xml I_f54adea1-a196-3a66-8c50-e1415fbd0429_1_R/BasicLTI.xml I_f558df97-0611-3e67-b78c-6a08cbafb5e1_1_R/BasicLTI.xml I_f558df97-0611-3e67-b78c-6a08cbafb5e1_3_R/BasicLTI.xml I_f564f3d6-1b90-3d10-a054-bbc17707ec03_R/BasicLTI.xml I_f5709037-995a-34b3-a846-15b1d1ac0f69_1_R/BasicLTI.xml I_f577ec3d-2a91-3fa4-9f04-e8f2dc2272cb_1_R/BasicLTI.xml I_f582a2fd-64d8-33dd-8662-e4223ac90c1c_R/BasicLTI.xml I_f585ddf1-87ad-3004-a301-303505433ffd_R/BasicLTI.xml I_f597325f-0db2-384e-9455-c26d01eafd47_1_R/BasicLTI.xml I_f5b941d8-61e6-362b-abf2-1153106917be_R/BasicLTI.xml I_f5baa3e3-ad5c-3f83-9ffe-f9fce60714b2_1_R/BasicLTI.xml I_f5bb59c6-ba31-38ad-b244-6ec1dc09b28b_1_R/BasicLTI.xml I_f5bc22ca-ca04-3067-a6d0-c1c9b8bf2f9d_1_R/BasicLTI.xml I_f5c577b6-d96e-3d20-9b49-70396bf040da_1_R/BasicLTI.xml I_f5d75df4-7079-34b6-919e-b12821517d4c_R/BasicLTI.xml I_f5f87cf1-7017-30dd-93f5-d54caa8359c3_1_R/BasicLTI.xml I_f60e0dd5-a4fe-3872-874d-0d7852589d57_1_R/BasicLTI.xml I_f61f49b2-b85d-30b3-91c5-10ce1bf9d752_R/BasicLTI.xml I_f62ed523-6deb-3f0a-a119-faf23d263e11_R/BasicLTI.xml I_f637c3b8-a981-3a1c-8753-f773a6184cee_1_R/BasicLTI.xml I_f643b6a1-1442-3a19-bf34-c2c761545610_R/BasicLTI.xml I_f657bb8e-5e1e-3c01-8905-28a149867bcd_1_R/BasicLTI.xml I_f682515b-4334-3531-9414-bb959268e120_R/BasicLTI.xml I_f68ea711-fc32-3cd6-8e2f-05dc047d62d3_1_R/BasicLTI.xml I_f690dc6d-1c8c-3d11-b66e-57e0329305e9_R/BasicLTI.xml I_f6b398a3-4878-3307-8d2d-f951efae2f09_1_R/BasicLTI.xml I_f6bdf180-55b1-3f73-b942-4271c6c61cdb_R/BasicLTI.xml I_f6c889a1-d2c6-3328-9861-c9cf38d4478b_1_R/BasicLTI.xml I_f6fb28d8-9384-3d70-8a5c-56f260c6f571_R/BasicLTI.xml I_f707ac93-95b2-3c37-9fa2-45a3c7b0b23a_1_R/BasicLTI.xml I_f71b2bdd-85e9-3129-90a7-1941cc431a4b_R/BasicLTI.xml I_f73711a4-92a4-376c-8102-a45bf06c3470_1_R/BasicLTI.xml I_f752dcf4-a5e8-3786-8566-bb86a895e2f6_1_R/BasicLTI.xml I_f753ed7b-ab45-33d3-967d-c96f9ef5df41_1_R/BasicLTI.xml I_f76d1ae0-3594-3d3a-9521-a4173db0eef0_1_R/BasicLTI.xml I_f777913b-b6bc-305e-bd0e-40b3353111e5_1_R/BasicLTI.xml I_f777913b-b6bc-305e-bd0e-40b3353111e5_3_R/BasicLTI.xml I_f79ec534-64cf-3685-92b7-d1e60f7e690f_1_R/BasicLTI.xml I_f7c1bb4c-dc38-30a3-ac5d-4c69108b4d48_1_R/BasicLTI.xml I_f7e10556-1d8b-3f61-82d0-0e1140109d59_1_R/BasicLTI.xml I_f7e6b42d-d036-3ded-a339-90272b4d1134_1_R/BasicLTI.xml I_f7e6b42d-d036-3ded-a339-90272b4d1134_3_R/BasicLTI.xml I_f7e6b42d-d036-3ded-a339-90272b4d1134_5_R/BasicLTI.xml I_f7e6b42d-d036-3ded-a339-90272b4d1134_7_R/BasicLTI.xml I_f7f2371f-d595-3767-84bc-7f02a053e664_1_R/BasicLTI.xml I_f80213d9-3638-3aba-a5ea-85ec18980183_1_R/BasicLTI.xml I_f8290861-aa05-3e35-a073-769c33aaa37e_1_R/BasicLTI.xml I_f82d0705-b70c-3a28-9613-358eff35fe6d_1_R/BasicLTI.xml I_f8421a0f-7b0d-3ed8-b4d8-e7ba9933fb18_1_R/BasicLTI.xml I_f8595639-ea73-301a-9c1c-97cd40d409d4_R/BasicLTI.xml I_f86a3274-64ee-3b31-b3d6-a0c43d39b42b_R/BasicLTI.xml I_f86f80cf-1f3a-3a7b-9e64-91fba785b3d5_R/BasicLTI.xml I_f87b902f-75fc-35b8-b60f-0a90136342e7_1_R/BasicLTI.xml I_f89ac5c4-2699-35b3-89e6-c8765eda7826_1_R/BasicLTI.xml I_f8a3eb64-40d2-36d2-a5dd-e2658b3d1282_R/BasicLTI.xml I_f8c6a0e9-2279-3912-ad93-76e3bcf2a1fd_1_R/BasicLTI.xml I_f8cebf99-668f-3d7b-9b69-88d18542a48b_1_R/BasicLTI.xml I_f8dc9453-17f4-34be-81e2-4726c45f1f2f_1_R/BasicLTI.xml I_f8dc9453-17f4-34be-81e2-4726c45f1f2f_3_R/BasicLTI.xml I_f8e06cea-0b5c-3e1b-932b-dc54261c15aa_1_R/BasicLTI.xml I_f8f8d036-73d5-3080-a00d-a6b375dd5fef_1_R/BasicLTI.xml I_f8fc9d61-2519-36b4-bdc8-3fedda283e02_R/BasicLTI.xml I_f91a9ae1-dc40-3cb3-8601-fec96f8af457_1_R/BasicLTI.xml I_f945f07c-d4d3-3bc8-a7ad-995ac41939da_1_R/BasicLTI.xml I_f973456d-e768-3b94-aee5-0f3d9aab38d6_R/BasicLTI.xml I_f97bed08-1c2e-3192-ab4f-cf9896d2fdcd_1_R/BasicLTI.xml I_f9bf6f08-f39c-3565-961a-70e5bcf0d9c9_1_R/BasicLTI.xml I_f9d7f184-4af6-3e55-a2b0-75b44517e410_R/BasicLTI.xml I_f9db3a7b-354b-36e6-90f7-94fdefb355ff_1_R/BasicLTI.xml I_f9e1099e-2814-324a-8f22-7a45f56f2b5c_R/BasicLTI.xml I_fa17282d-ad2e-3d90-95d7-b07fe96e8358_R/BasicLTI.xml I_fa31e541-9468-3413-89a0-9e9c48155b56_R/BasicLTI.xml I_fa3807f8-9d6c-3493-8785-00efbcd19442_1_R/BasicLTI.xml I_fa4aec29-dc6f-3071-bb7a-f28e18532e1f_R/BasicLTI.xml I_fa529f01-bd9d-3b2a-8fe9-5c11be657540_1_R/BasicLTI.xml I_fa6e0e27-2806-3e68-b9b0-87b5c33b9cab_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_11_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_13_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_1_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_3_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_5_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_7_R/BasicLTI.xml I_fa6f0244-4a01-333c-bb19-fb9d57976a59_9_R/BasicLTI.xml I_fa73531a-6eb0-3d75-8375-b05a3f1d9a04_R/BasicLTI.xml I_fa7aa1ed-7c71-35f7-875a-cc22ed7aae1c_1_R/BasicLTI.xml I_fa7aa1ed-7c71-35f7-875a-cc22ed7aae1c_3_R/BasicLTI.xml I_fa7f09a8-2565-3976-a204-f342ba1c96ba_1_R/BasicLTI.xml I_fa80654d-1bc5-3d48-bb27-c3aec6be9712_1_R/BasicLTI.xml I_fa9c24da-8d77-3ff0-bd10-94db58a745f7_1_R/BasicLTI.xml I_faedd889-72e0-38ce-8d28-b8c144a8634e_1_R/BasicLTI.xml I_faedd889-72e0-38ce-8d28-b8c144a8634e_3_R/BasicLTI.xml I_faedd889-72e0-38ce-8d28-b8c144a8634e_5_R/BasicLTI.xml I_faee5e0c-3a2e-39d3-91a6-6fd75da23f3a_1_R/BasicLTI.xml I_fafe398f-ce89-3740-a930-9a6f29738503_1_R/BasicLTI.xml I_fb0aff0e-b347-3e11-8a25-762d7791e634_R/BasicLTI.xml I_fb362cc1-9672-3029-8c57-54782d3470b5_R/BasicLTI.xml I_fb3a2982-7a6d-3e60-a15c-91f551b5e4f4_R/BasicLTI.xml I_fb50657a-44f5-3c99-a513-2712bbc469c5_R/BasicLTI.xml I_fb66e958-a847-3fad-86d9-f507ddb4fe43_R/BasicLTI.xml I_fb9b97fb-6f51-38e9-a488-45ac5e51be15_1_R/BasicLTI.xml I_fbb7b36e-988f-3d6c-93a4-cd5df6b64f0a_1_R/BasicLTI.xml I_fbbcad61-011d-3084-bfd5-faae6dfddd01_1_R/BasicLTI.xml I_fbd2da49-a1eb-3807-8028-3aa98c795d25_1_R/BasicLTI.xml I_fbef2cd4-d93e-3f77-9961-2d61e8d590d2_1_R/BasicLTI.xml I_fc1da22b-c858-333e-8417-0aa1e1ca9a1c_R/BasicLTI.xml I_fc3484b2-231d-3678-a600-3e45aba74ca7_R/BasicLTI.xml I_fc43e129-0b7d-327f-8ebc-0ed1bb0e26e7_R/BasicLTI.xml I_fc4ca8d4-3b3f-32f4-8eb8-d0070422c782_1_R/BasicLTI.xml I_fc618b45-581b-3777-bb20-792a7d85f52b_1_R/BasicLTI.xml I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_1_R/BasicLTI.xml I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_3_R/BasicLTI.xml I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_5_R/BasicLTI.xml I_fc6d55c0-83d2-3baa-a6d7-76daa527580a_7_R/BasicLTI.xml I_fc795497-8526-3d18-9630-a81c627433c0_1_R/BasicLTI.xml I_fc8ce413-df07-322b-b8b7-ecb906e8696e_R/BasicLTI.xml I_fc9657ff-88db-36ab-baef-8f6e663c7d56_R/BasicLTI.xml I_fcae4165-de06-3683-aaf0-6d05d2e4d5be_R/BasicLTI.xml I_fccc0440-567d-3869-a93d-98680934d06d_R/BasicLTI.xml I_fce469e3-e0dd-3d9a-872a-5b823ca67524_1_R/BasicLTI.xml I_fce469e3-e0dd-3d9a-872a-5b823ca67524_3_R/BasicLTI.xml I_fce469e3-e0dd-3d9a-872a-5b823ca67524_5_R/BasicLTI.xml I_fce469e3-e0dd-3d9a-872a-5b823ca67524_7_R/BasicLTI.xml I_fceaa64a-a90f-3dee-a435-dce5f5060f4c_1_R/BasicLTI.xml I_fcf02b38-a153-3ae6-9a4a-bd7fcc39e491_R/BasicLTI.xml I_fd06e78c-d6d3-3c3d-ae5c-46fdf880776c_1_R/BasicLTI.xml I_fd07e666-6455-3c37-8ffa-8099fe6fd70b_R/BasicLTI.xml I_fd2cebbc-57a9-3ad5-ae93-9e623b6e77ce_1_R/BasicLTI.xml I_fd33e172-92e4-3ed2-8af0-9186bcba65d8_1_R/BasicLTI.xml I_fd3b9cdb-bfa1-378f-8abb-3dbd92155e70_1_R/BasicLTI.xml I_fd598991-e252-399b-b157-0ce36664c235_R/BasicLTI.xml I_fd668180-e69b-3f74-b444-98dde753debb_1_R/BasicLTI.xml I_fd7ee9e3-2c69-3f78-b426-1055cf228aa0_R/BasicLTI.xml I_fd853334-2227-38a3-876e-261f5cdb4f64_1_R/BasicLTI.xml I_fdb728a4-43af-3434-9d39-a798cbe86e70_R/BasicLTI.xml I_fdcb85f6-5658-3786-b041-bd6f38ac6731_1_R/BasicLTI.xml I_fdf24ca4-d3be-3000-a270-391ddeb6d1e4_R/BasicLTI.xml I_fdf30f45-e230-34f1-8e45-fcaaf0d5011a_R/BasicLTI.xml I_fdf32da3-ce26-3615-937f-878d21996bd8_R/BasicLTI.xml I_fe026fa8-18a0-32c8-9ec2-394f40c68838_1_R/BasicLTI.xml I_fe09eba3-e170-303e-a96d-3856599565ee_R/BasicLTI.xml I_fe18fb72-53c4-3cbf-a3ec-d366892745b7_1_R/BasicLTI.xml I_fe198dee-f250-388c-b603-dab76ba2816b_R/BasicLTI.xml I_fe2419c5-74ab-3567-933d-a5ddafbb65da_1_R/BasicLTI.xml I_fe354d2e-15ad-3d14-a359-c3551e5b80da_1_R/BasicLTI.xml I_fe391a3d-f5e9-305d-b02a-40c1dc08f4f5_1_R/BasicLTI.xml I_fe5464c2-5b0f-3731-90fb-53fec4c059b9_R/BasicLTI.xml I_fe5d2997-2408-3ca1-8aef-de160d65460a_1_R/BasicLTI.xml I_fe79a959-9c74-33ff-aac2-340928b451fc_R/BasicLTI.xml I_fe87f7d3-7d04-3c91-a1e9-67e3310c2229_1_R/BasicLTI.xml I_fe97d97f-d957-36fc-9349-8ebfe25b2f0a_1_R/BasicLTI.xml I_fecdf7fb-af2e-3009-aae6-ceb62420c99e_1_R/BasicLTI.xml I_fee7d3cc-af6b-3a3d-a879-78c7b98b94d9_R/BasicLTI.xml I_ff01324f-c91d-34d2-9ddc-a91431e25fdf_1_R/BasicLTI.xml I_ff10693d-35e5-3093-89fc-d9be32f5d8e6_R/BasicLTI.xml I_ff3ae573-f55f-3af6-80b9-732ccc6f7aa1_1_R/BasicLTI.xml I_ff472eb6-13aa-3d49-9cbe-184329b9f388_1_R/BasicLTI.xml I_ff4db4b1-2aa6-345d-89de-1170043bd7a5_1_R/BasicLTI.xml I_ff6f1256-b63f-3b09-9ad4-d9f1e5a942d0_1_R/BasicLTI.xml I_ff6f1256-b63f-3b09-9ad4-d9f1e5a942d0_3_R/BasicLTI.xml I_ff6f1256-b63f-3b09-9ad4-d9f1e5a942d0_5_R/BasicLTI.xml I_ff6f1256-b63f-3b09-9ad4-d9f1e5a942d0_7_R/BasicLTI.xml I_ff7139b1-5d36-3489-b4f1-48345fee32bd_1_R/BasicLTI.xml I_ff7139b1-5d36-3489-b4f1-48345fee32bd_3_R/BasicLTI.xml I_ff7139b1-5d36-3489-b4f1-48345fee32bd_5_R/BasicLTI.xml I_ff8eb85b-8637-3b61-97be-222ad769ec46_1_R/BasicLTI.xml I_ff8f052c-9f8b-32d8-8a78-672fa0bedaa0_R/BasicLTI.xml I_ff93f0da-49e6-33c3-aae7-ff1126d37c98_1_R/BasicLTI.xml I_ff983e3a-cc32-39fb-b5c4-2a3af5b05f62_R/BasicLTI.xml I_ffa15c3a-fb95-38ad-aa03-93bffe97adb4_R/BasicLTI.xml I_ffa2669d-4285-3366-aa5e-6957e13c0670_1_R/BasicLTI.xml I_ffa6d6d8-b6c0-3c6b-93db-de53371e8feb_R/BasicLTI.xml I_ffab2a40-8c03-3d23-bb43-d963487090df_R/BasicLTI.xml I_ffb8531d-2eca-3dee-b727-759ce03b2a5c_1_R/BasicLTI.xml I_ffe21d40-7edd-30b8-b0aa-53c79ba726eb_1_R/BasicLTI.xml I_ffe2779d-833b-33fa-9488-35e1aa3e0247_R/BasicLTI.xml I_ffe9a8ae-5a91-361e-ae0b-add549c6700c_R/BasicLTI.xml I_fff20a3c-1d2d-3856-908f-e715d11532e3_R/BasicLTI.xml I_fff38cfa-7b69-3116-a805-bdf22b8d0cc8_R/BasicLTI.xml I_fffbff91-1b3f-3f5b-a656-131c0599f70c_R/BasicLTI.xml Title: enVision Integrated Mathematics II Description: enVision Integrated Mathematics II Interactive Student Edition: Realize Reader: Mathematics II Beginning-of-Year Assessment (PDF) Beginning-of-Year Assessment Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Write and graph linear equations using standard form. Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Write and graph linear equations using point-slope form. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Identify and describe arithmetic sequences. Rewrite and use literal equations to solve problems. Write and solve compound inequalities. Solve and graph inequalities. Write and graph linear equations using slope-intercept form. Identify, evaluate, and graph linear functions. Recognize the key features of exponential functions. Perform, analyze, and use transformations of exponential functions. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. applying slope to verify and determine whether lines are parallel or perpendicular; Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Exponents and Roots Interactive Student Edition: Realize Reader: Topic 1 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 5-1: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. 5-3: MathXL for School: Enrichment Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 3-2: Ex 2: Write a Linear Function Rule & Try It! Curriculum Standards: Identify, evaluate, and graph linear functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship or two input-output pairs (include reading these from a table). Instructional Note: In constructing linear functions, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.) Identify, evaluate, and graph linear functions. 2-3: Virtual Nerd™: How Do You Use X- and Y-Intercepts To Graph a Line In Standard Form? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.) Instructional Note: Address this standard before discussing exponential functions with continuous domains. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Understand and apply the relationship of rational exponents to integer exponents and radicals to solve problems. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. 5-1: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. 2-1: Ex 2: Write an Equation from a Graph & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. 5-1: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. 5-3: Ex 2: Exponential Models of Growth & Try It! Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 2-1: Virtual Nerd™: How Do You Write an Equation of a Line in Slope-Intercept Form if You Have a Graph? Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. 3-2: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-1: Ex 1: Graph a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 5-3: Virtual Nerd™: What is Exponential Growth? Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. Topic 1: Readiness Assessment (PDF) Curriculum Standards: Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. Rewrite and use literal equations to solve problems. Reason about operations with real numbers. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Topic 1: Readiness Assessment Curriculum Standards: Reason about operations with real numbers. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. Write equivalent radical expressions. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Identify, evaluate, and graph linear functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Topic 1: enVision STEM Project Topic 1: enVision STEM Project (PDF) Topic 1: enVision STEM Video Topic 1: enVision STEM Masters (PDF) Operations on Real Numbers Interactive Student Edition: Realize Reader: Lesson 1-1 Curriculum Standards: Reason about operations with real numbers. Explore 1-1: Critique & Explain Curriculum Standards: Reason about operations with real numbers. Understand and Apply 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Additional Example 1 Curriculum Standards: Reason about operations with real numbers. 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 3: Operations With Rational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 4: Operations With Rational and Irrational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. 1-1: Additional Example 4 with Try Another One Curriculum Standards: Reason about operations with real numbers. 1-1: Concept Summary Curriculum Standards: Reason about operations with real numbers. 1-1: Do You Understand? Curriculum Standards: Reason about operations with real numbers. 1-1: Do You Know How? Curriculum Standards: Reason about operations with real numbers. Practice and Problem Solving 1-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Reason about operations with real numbers. Assess & Differentiate 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 4: Operations With Rational and Irrational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. 1-1: Virtual Nerd™: What's an Irrational Number? Curriculum Standards: Reason about operations with real numbers. 1-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Reason about operations with real numbers. 1-1: MathXL for School: Enrichment Curriculum Standards: Reason about operations with real numbers. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Lesson Quiz (PDF) Curriculum Standards: Reason about operations with real numbers. 1-1: Lesson Quiz Curriculum Standards: Reason about operations with real numbers. 1-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Reason about operations with real numbers. 1-1: Reteach to Build Understanding (PDF) Curriculum Standards: Reason about operations with real numbers. 1-1: MathXL for School: Additional Practice Curriculum Standards: Reason about operations with real numbers. 1-1: Additional Practice (PDF) Curriculum Standards: Reason about operations with real numbers. 1-1: MathXL for School: Enrichment Curriculum Standards: Reason about operations with real numbers. 1-1: Enrichment (PDF) Curriculum Standards: Reason about operations with real numbers. 1-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Reason about operations with real numbers. 1-1: Virtual Nerd™: What's an Irrational Number? Curriculum Standards: Reason about operations with real numbers. Rational Exponents and Properties of Exponents Interactive Student Edition: Realize Reader: Lesson 1-2 Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Explore 1-2: Critique & Explain Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Understand and Apply 1-2: Ex 1: Write Radicals Using Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 2: Use the Product of Powers Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Additional Example 2 with Try Another One Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 3: Use the Power of a Power Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Additional Example 4 Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 5: Use the Quotient of Powers Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Concept Summary Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Do You Understand? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Do You Know How? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Practice and Problem Solving 1-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Assess & Differentiate 1-2: Ex 1: Write Radicals Using Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 3: Use the Power of a Power Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Lesson Quiz (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Lesson Quiz Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Reteach to Build Understanding (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Additional Practice Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Additional Practice (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Enrichment (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are the Properties of Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Exponential Models Interactive Student Edition: Realize Reader: Lesson 1-3 Explore 1-3: Explore & Reason Understand and Apply 1-3: Ex 1: Rewrite an Exponential Function to Identify a Rate & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: Additional Example 1 with Try Another One Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: Concept: Compound Interest 1-3: Ex 2: Understand Compound Interest & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: Ex 3: Understanding Continuously Compounded Interest & Try-It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Ex 4: Find Continuously Compounded Interest & Try-It! Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: Additional Example 4 Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: Ex 5: Use Two Points to Find an Exponential Model & Try-It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Ex 6: Use Regression to Find an Exponential Model & Try-It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Concept Summary 1-3: Do You Understand? Curriculum Standards: Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Do You Know How? Curriculum Standards: Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Practice and Problem Solving 1-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Assess & Differentiate 1-3: Ex 2: Understand Compound Interest & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: Ex 4: Find Continuously Compounded Interest & Try-It! Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: Additional Example 4 Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: MathXL for School: Enrichment Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: Ex 3: Understanding Continuously Compounded Interest & Try-It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Ex 5: Use Two Points to Find an Exponential Model & Try-It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Lesson Quiz (PDF) 1-3: Lesson Quiz Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: Reteach to Build Understanding (PDF) Curriculum Standards: Write exponential models in different ways to solve problems. 1-3: MathXL for School: Additional Practice Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Additional Practice (PDF) 1-3: MathXL for School: Enrichment Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. 1-3: Enrichment (PDF) 1-3: Mathematical Literacy and Vocabulary (PDF) 1-3: Virtual Nerd™: How Do You Use the Formula for Compound Interest? Curriculum Standards: Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-3: Virtual Nerd™: What is a Natural Base Exponential Function? Curriculum Standards: Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Mathematical Modeling in 3 Acts: Edgy Tiles Topic 1: Edgy Tiles - Act 1 Video With Questions Topic 1: Edgy Tiles - Act 2 Content Topic 1: Edgy Tiles - Act 2 Questions Topic 1: Edgy Tiles - Act 3 Video Topic 1: Edgy Tiles - Act 3 Questions The Square Root Function Interactive Student Edition: Realize Reader: Lesson 1-4 Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. Write and graph quadratic functions in standard form. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Analyze functions that include absolute value expressions. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Explore 1-4: Explore & Reason Understand and Apply 1-4: Ex 1: Key Features of the Square Root Function & Try It! Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. 1-4: Additional Example 1 with Try Another One Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. 1-4: Ex 2: Translations of the Square Root Function & Try It! Curriculum Standards: Describe the key features of the square root function. 1-4: Additional Example 2 Curriculum Standards: Describe the key features of the square root function. 1-4: Ex 3: Rate of Change of the Square Root Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: Ex 4: Compare Functions & Try It! Curriculum Standards: Describe the key features of the square root function. Write and graph quadratic functions in standard form. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. 1-4: Concept Summary Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. Write and graph quadratic functions in standard form. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Analyze functions that include absolute value expressions. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: Do You Understand? Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. Write and graph quadratic functions in standard form. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Analyze functions that include absolute value expressions. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: Do You Know How? Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. Write and graph quadratic functions in standard form. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Analyze functions that include absolute value expressions. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Practice and Problem Solving 1-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and transform radical functions. Assess & Differentiate 1-4: Ex 2: Translations of the Square Root Function & Try It! Curriculum Standards: Describe the key features of the square root function. 1-4: Additional Example 2 Curriculum Standards: Describe the key features of the square root function. 1-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-4: MathXL for School: Enrichment Curriculum Standards: Describe the key features of the square root function. 1-4: Ex 3: Rate of Change of the Square Root Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: Virtual Nerd™: How Do You Graph a Square Root Function Using a Table? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: Ex 1: Key Features of the Square Root Function & Try It! Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. 1-4: Virtual Nerd™: What Does the Parent Function Graph of a Square Root Function Look Like? Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. 1-4: Lesson Quiz (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and transform radical functions. 1-4: Lesson Quiz Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and transform radical functions. 1-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-4: Reteach to Build Understanding (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-4: MathXL for School: Additional Practice Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and transform radical functions. 1-4: Additional Practice (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and transform radical functions. 1-4: MathXL for School: Enrichment Curriculum Standards: Describe the key features of the square root function. 1-4: Enrichment (PDF) Curriculum Standards: Describe the key features of the square root function. 1-4: Mathematical Literacy and Vocabulary (PDF) 1-4: Virtual Nerd™: What Does the Parent Function Graph of a Square Root Function Look Like? Curriculum Standards: Describe the key features of the square root function. Graph and transform radical functions. 1-4: Virtual Nerd™: How Do You Graph a Square Root Function Using a Table? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. The Cube Root Function Interactive Student Edition: Realize Reader: Lesson 1-5 Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Explore 1-5: Critique & Explain Understand and Apply 1-5: Ex 1: Key Features of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. 1-5: Ex 2: Translations of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Additional Example 2 with Try Another One Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Ex 3: Model a Problem Using the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. 1-5: Ex 4: Compare Rates of Change of a Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 1-5: Additional Example 4 Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Ex 5: Compare Rates of Change of Two Functions & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Concept Summary Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Do You Understand? Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Do You Know How? Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Practice and Problem Solving 1-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Assess & Differentiate 1-5: Ex 1: Key Features of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. 1-5: Ex 4: Compare Rates of Change of a Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 1-5: Virtual Nerd™: How Do You Graph a Cube Root Function Using a Table? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 1-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Ex 2: Translations of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ?(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Additional Example 2 with Try Another One Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: MathXL for School: Enrichment Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Ex 3: Model a Problem Using the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. 1-5: Lesson Quiz (PDF) Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Lesson Quiz Curriculum Standards: Identify the key features of the cube root function. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Reteach to Build Understanding (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: MathXL for School: Additional Practice Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: Additional Practice (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. 1-5: MathXL for School: Enrichment Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-5: Enrichment (PDF) Curriculum Standards: Identify the key features of the cube root function. 1-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Identify the key features of the cube root function. 1-5: Virtual Nerd™: How Do You Graph a Cube Root Function Using a Table? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. Topic 1: MathXL for School: Topic Review Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Graph solutions to linear inequalities in two variables. Solve systems of linear equations using the elimination method. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Graph and solve a system of linear inequalities. Solve linear-quadratic systems. Topic 1: Performance Assessment Form A (PDF) Topic 1: Performance Assessment Form A Curriculum Standards: Graph solutions to linear inequalities in two variables. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Topic 1: Performance Assessment Form B Topic 1: Performance Assessment Form B (PDF) 5-2: Ex 3: Understanding Continuously Compounded Interest & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. Solve exponential and logarithmic equations. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-5: Ex 2: Translations of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 5-3: MathXL for School: Enrichment Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 5-2: Ex 5: Use Two Points to Find an Exponential Model & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. Solve exponential and logarithmic equations. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-5: MathXL for School: Enrichment Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-1: Ex 3: Operations With Rational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-5: Additional Example 2 with Try Another One Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 5-3: Ex 2: Exponential Models of Growth & Try It! Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 5-3: Virtual Nerd™: What is Exponential Growth? Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. Topic 1: Assessment Form A (PDF) Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Solve linear-quadratic systems. Solve systems of linear equations using the elimination method. Topic 1: Assessment Form A Curriculum Standards: Reason about operations with real numbers. Identify the key features of the cube root function. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. Write equivalent radical expressions. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Topic 1: Assessment Form B Topic 1: Assessment Form B (PDF) 5-2: Ex 3: Understanding Continuously Compounded Interest & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. Solve exponential and logarithmic equations. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-5: Ex 2: Translations of the Cube Root Function & Try It! Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??((?), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????((?), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 5-3: MathXL for School: Enrichment Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 5-2: Ex 5: Use Two Points to Find an Exponential Model & Try-It! Curriculum Standards: Write exponential models in different ways to solve problems. Solve exponential and logarithmic equations. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 1-5: MathXL for School: Enrichment Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ) (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-1: Ex 3: Operations With Rational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-5: Additional Example 2 with Try Another One Curriculum Standards: Identify the key features of the cube root function. Identify the key features of the cube root function. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 5-3: Ex 2: Exponential Models of Growth & Try It! Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 5-3: Virtual Nerd™: What is Exponential Growth? Curriculum Standards: Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Graph exponential and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline and amplitude. Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Interpret the parameters in a linear or exponential function in terms of a context. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Interpret the parameters in a linear or exponential function in terms of the context. Topic 1: Assessment Form C Curriculum Standards: Reason about operations with real numbers. Identify the key features of the cube root function. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. Write equivalent radical expressions. Use exponential functions to model situations and make predictions. Write exponential models in different ways to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Polynomials and Factoring 1-1: Additional Example 1 Curriculum Standards: Reason about operations with real numbers. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-2: Additional Example 2 with Try Another One Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Additional Example 4 Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 3: Use the Power of a Power Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 5: Use the Quotient of Powers Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are the Properties of Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Topic 2: Readiness Assessment (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Topic 2: Readiness Assessment Curriculum Standards: Reason about operations with real numbers. Identify, evaluate, and graph linear functions. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Topic 2: enVision STEM Project Topic 2: enVision STEM Project (PDF) Topic 2: enVision STEM Video Topic 2: enVision STEM Masters (PDF) Adding and Subtracting Polynomials Interactive Student Edition: Realize Reader: Lesson 2-1 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Explore 2-1: Explore & Reason Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Understand and Apply 2-1: Ex 1: Understand Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Ex 2: Write Polynomials in Standard Form & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Ex 3: Add and Subtract Monomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Ex 4: Add Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 5: Subtract Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Additional Example 5 with Try Another One Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Ex 6: Apply Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Additional Example 6 Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Concept Summary Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Do You Understand? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Do You Know How? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Practice and Problem Solving 2-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Assess & Differentiate 2-1: Ex 4: Add Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Additional Example 6 Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Add, subtract, multiply, divide, and simplify polynomial and rational expressions. 2-1: MathXL for School: Enrichment Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 5: Subtract Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Virtual Nerd™: How Do You Subtract Polynomials? Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 1: Understand Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Virtual Nerd™: What's the Standard Form of a Polynomial? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Lesson Quiz (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Lesson Quiz Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Add, subtract, multiply, divide, and simplify polynomial and rational expressions. 2-1: Reteach to Build Understanding (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: MathXL for School: Additional Practice Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Additional Practice (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: MathXL for School: Enrichment Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Enrichment (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Virtual Nerd™: What's the Standard Form of a Polynomial? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-1: Virtual Nerd™: How Do You Subtract Polynomials? Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Multiplying Polynomials Interactive Student Edition: Realize Reader: Lesson 2-2 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Explore 2-2: Model & Discuss Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Understand and Apply 2-2: Ex 1: Multiply a Monomial and a Trinomial & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Ex 2: Use a Table to Find the Product of Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Additional Example 2 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Ex 3: Multiply Binomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Ex 4: Multiply a Trinomial and a Binomial & Try It! Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: Ex 5: Closure and Multiplication & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Ex 6: Apply Multiplication of Binomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Additional Example 6 with Try Another One Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Concept Summary Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Do You Understand? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Do You Know How? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Practice and Problem Solving 2-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Assess & Differentiate 2-2: Ex 4: Multiply a Trinomial and a Binomial & Try It! Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: Virtual Nerd™: How Do You Multiply a Binomial and a Trinomial Using the Grid Method? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: MathXL for School: Enrichment Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Lesson Quiz (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Lesson Quiz Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: Reteach to Build Understanding (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: MathXL for School: Additional Practice Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Additional Practice (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: MathXL for School: Enrichment Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Enrichment (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Virtual Nerd™: How Do You Multiply a Binomial and a Trinomial Using the Grid Method? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Virtual Nerd™: How Do You Multiply Binomials Using the Distributive Property? Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Multiplying Special Cases Interactive Student Edition: Realize Reader: Lesson 2-3 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Explore 2-3: Explore & Reason Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Understand and Apply 2-3: Ex 1: Determine the Square of a Binomial & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Additional Example 1C with Try Another One Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Ex 2: Find the Product of a Sum and a Difference & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Ex 3: Apply the Square of a Binomial & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Additional Example 3 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Concept Summary Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Do You Understand? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Do You Know How? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Practice and Problem Solving 2-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Assess & Differentiate 2-3: Ex 2: Find the Product of a Sum and a Difference & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Virtual Nerd™: What's the Formula for the Product of a Sum and a Difference? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: MathXL for School: Enrichment Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-3: Lesson Quiz (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Lesson Quiz Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Reteach to Build Understanding (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: MathXL for School: Additional Practice Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Additional Practice (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: MathXL for School: Enrichment Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-3: Enrichment (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Virtual Nerd™: What's the Formula for the Product of a Sum and a Difference? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-3: Virtual Nerd™: How Do You Use the Formula for the Product of a Sum and a Difference? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Factoring Polynomials Interactive Student Edition: Realize Reader: Lesson 2-4 Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Explore 2-4: Model & Discuss Understand and Apply 2-4: Ex 1: Find the Greatest Common Factor & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Ex 2: Factor Out the Greatest Common Factor & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Additional Example 2 with Try Another One Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Ex 3: Factor a Polynomial Model & Try It! Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Additional Example 3A Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Concept Summary Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Do You Understand? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Do You Know How? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Practice and Problem Solving 2-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Assess & Differentiate 2-4: Ex 3: Factor a Polynomial Model & Try It! Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Virtual Nerd™: How Do You Factor the Greatest Common Factor Out of Monomials? Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: MathXL for School: Enrichment Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Lesson Quiz (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Lesson Quiz Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Reteach to Build Understanding (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: MathXL for School: Additional Practice Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Additional Practice (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: MathXL for School: Enrichment Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Enrichment (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-4: Mathematical Literacy and Vocabulary (PDF) 2-4: Virtual Nerd™: How Do You Factor the Greatest Common Factor Out of Monomials? Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: Virtual Nerd™: How Do You Factor the Greatest Common Factor out of a Polynomial? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factoring ??² + ???? + ?? Interactive Student Edition: Realize Reader: Lesson 2-5 Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Explore 2-5: Explore & Reason Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Understand and Apply 2-5: Ex 1: Understand Factoring a Trinomial & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: Additional Example 1B Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Ex 2: Factor x² + ???? + c , When ?? < 0 and ?? > 0 & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Ex 3: Factor x² + ???? + c , When ?? < 0 & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Additional Example 3 with Try Another One Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Ex 4: Factor ?? Trinomial With Two Variables & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Ex 5: Apply Factoring Trinomials & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Concept Summary Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Do You Understand? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Do You Know How? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Practice and Problem Solving 2-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Assess & Differentiate 2-5: Ex 1: Understand Factoring a Trinomial & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: Virtual Nerd™: How Do You Factor a Trinomial? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: Lesson Quiz (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Lesson Quiz Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: Reteach to Build Understanding (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: MathXL for School: Additional Practice Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-5: Additional Practice (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: MathXL for School: Enrichment Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-5: Enrichment (PDF) 2-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: Virtual Nerd™: How Do You Factor a Trinomial? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-5: Virtual Nerd™: How Do You Factor a Perfect Square Trinomial by Guess and Check? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Mathematical Modeling in 3 Acts: Who's Right? Topic 2: Who's Right? - Act 1 Video with Questions Topic 2: Who's Right? - Act 2 Content Topic 2: Who's Right? - Act 2 Questions Topic 2: Who's Right? - Act 3 Video Topic 2: Who's Right? - Act 3 Questions Factoring ????² + ???? + ?? Interactive Student Edition: Realize Reader: Lesson 2-6 Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Explore 2-6: Explore & Reason Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Understand and Apply 2-6: Ex 1: Factor Out a Common Factor & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Additional Example 1 with Try Another One Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Ex 2: Understand Factoring by Grouping & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Additional Example 2 Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Ex 3: Factor ?? Trinomial Using Substitution & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Concept Summary Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Do You Understand? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Do You Know How? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Practice and Problem Solving 2-6: MathXL for School: Practice and Problem Solving Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Assess & Differentiate 2-6: Ex 1: Factor Out a Common Factor & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Virtual Nerd™: How Do You Factor a Common Factor Out Of a Difference of Squares? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-6: Lesson Quiz (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-6: Lesson Quiz Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-6: Reteach to Build Understanding (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-6: MathXL for School: Additional Practice Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-6: Additional Practice (PDF) 2-6: Enrichment (PDF) 2-6: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Virtual Nerd™: How Do You Factor a Common Factor Out Of a Difference of Squares? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. 2-6: Virtual Nerd™: How Do You Factor a Polynomial by Guessing and Checking? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factoring Special Cases Interactive Student Edition: Realize Reader: Lesson 2-7 Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Explore 2-7: Critique & Explain Understand and Apply 2-7: Ex 1: Understand Factoring a Perfect Square & Try It! Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Ex 2: Factor to Find a Dimension & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Ex 3: Factor a Difference of Two Squares & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Additional Example 3 with Try Another One Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Ex 4: Factor Out a Common Factor & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Additional Example 4 Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Concept Summary Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Do You Understand? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Do You Know How? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Practice and Problem Solving 2-7: MathXL for School: Practice and Problem Solving Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Assess & Differentiate 2-7: Ex 3: Factor a Difference of Two Squares & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Additional Example 3 with Try Another One Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: MathXL for School: Enrichment Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Ex 1: Understand Factoring a Perfect Square & Try It! Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Virtual Nerd™: How Do You Determine if You Have a Perfect Square Trinomial? Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Ex 2: Factor to Find a Dimension & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Lesson Quiz (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Lesson Quiz Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Reteach to Build Understanding (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Additional Practice Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-7: Additional Practice (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Enrichment Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Enrichment (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: Mathematical Literacy and Vocabulary (PDF) 2-7: Virtual Nerd™: How Do You Determine if You Have a Perfect Square Trinomial? Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Virtual Nerd™: What is a Perfect Square Trinomial? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Topic 2: MathXL for School: Topic Review Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Topic 2: Performance Assessment Form A (PDF) Topic 2: Performance Assessment Form A Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Topic 2: Performance Assessment Form B Topic 2: Performance Assessment Form B (PDF) 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: MathXL for School: Enrichment Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: MathXL for School: Enrichment Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: Ex 3: Factor a Polynomial Model & Try It! Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Ex 3: Factor a Difference of Two Squares & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-5: Virtual Nerd™: How Do You Factor a Trinomial? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-3: MathXL for School: Enrichment Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 4: Add Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-5: Ex 1: Understand Factoring a Trinomial & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Add, subtract, multiply, divide, and simplify polynomial and rational expressions. 2-1: Virtual Nerd™: What's the Standard Form of a Polynomial? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-7: Ex 2: Factor to Find a Dimension & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Enrichment Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-2: Ex 4: Multiply a Trinomial and a Binomial & Try It! Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 5: Subtract Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-7: Ex 1: Understand Factoring a Perfect Square & Try It! Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Virtual Nerd™: How Do You Determine if You Have a Perfect Square Trinomial? Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: Additional Example 6 Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-7: Additional Example 3 with Try Another One Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-1: Ex 1: Understand Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Virtual Nerd™: How Do You Multiply Binomials Using the Distributive Property? Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: Virtual Nerd™: How Do You Factor the Greatest Common Factor Out of Monomials? Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: Virtual Nerd™: How Do You Subtract Polynomials? Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Topic 2: Assessment Form A (PDF) Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Topic 2: Assessment Form A Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Topic 2: Assessment Form B Topic 2: Assessment Form B (PDF) 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-4: MathXL for School: Enrichment Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: MathXL for School: Enrichment Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: Ex 3: Factor a Polynomial Model & Try It! Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Ex 3: Factor a Difference of Two Squares & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-5: Virtual Nerd™: How Do You Factor a Trinomial? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-3: MathXL for School: Enrichment Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 4: Add Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-5: Ex 1: Understand Factoring a Trinomial & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Add, subtract, multiply, divide, and simplify polynomial and rational expressions. 2-1: Virtual Nerd™: What's the Standard Form of a Polynomial? Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-7: Ex 2: Factor to Find a Dimension & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-7: MathXL for School: Enrichment Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-2: Ex 4: Multiply a Trinomial and a Binomial & Try It! Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-1: Ex 5: Subtract Polynomials & Try It! Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-7: Ex 1: Understand Factoring a Perfect Square & Try It! Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-7: Virtual Nerd™: How Do You Determine if You Have a Perfect Square Trinomial? Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: Additional Example 6 Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-7: Additional Example 3 with Try Another One Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 2-1: Ex 1: Understand Polynomials & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 2-2: Virtual Nerd™: How Do You Multiply Binomials Using the Distributive Property? Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-4: Virtual Nerd™: How Do You Factor the Greatest Common Factor Out of Monomials? Curriculum Standards: Factor a polynomial. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor a polynomial. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 2-1: Virtual Nerd™: How Do You Subtract Polynomials? Curriculum Standards: Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Combine like terms to simplify polynomials. Add, subtract, and multiply polynomials. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor special trinomials. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Factor special trinomials. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Topic 2: Assessment Form C Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 1-1: Ex 2: Compare and Order Real Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 1: Understand Sets and Subsets & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-1: Ex 3: Operations With Rational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 5-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve exponential and logarithmic equations. For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. Solve common and natural logarithmic equations using the properties of logarithms. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. For exponential models, express as a logarithm the solution to ???? to the ???? power = ?? where ??, ??, and ?? are numbers and the base ?? is 2, 10, or ??; evaluate the logarithm using technology. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve exponential and logarithmic equations. 5-7: Ex 2: Rewrite Exponential Equations Using Logarithms & Try-It! Curriculum Standards: Solve exponential and logarithmic equations. For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. Solve common and natural logarithmic equations using the properties of logarithms. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. For exponential models, express as a logarithm the solution to ???? to the ???? power = ?? where ??, ??, and ?? are numbers and the base ?? is 2, 10, or ??; evaluate the logarithm using technology. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve exponential and logarithmic equations. 5-7: Ex 1: Solve Exponential Equations Using a Common Base & Try It! Curriculum Standards: Solve exponential and logarithmic equations. For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. Solve common and natural logarithmic equations using the properties of logarithms. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. For exponential models, express as a logarithm the solution to ???? to the ???? power = ?? where ??, ??, and ?? are numbers and the base ?? is 2, 10, or ??; evaluate the logarithm using technology. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve exponential and logarithmic equations. 5-7: MathXL for School: Enrichment Curriculum Standards: Solve exponential and logarithmic equations. For exponential models, express as a logarithm the solution to a b to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Instructional Note: Consider extending this unit to include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that log xy = log x +log y. Solve common and natural logarithmic equations using the properties of logarithms. Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. For exponential models, express as a logarithm the solution to ???? to the ???? power = ?? where ??, ??, and ?? are numbers and the base ?? is 2, 10, or ??; evaluate the logarithm using technology. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve exponential and logarithmic equations. Benchmark Test 1 (PDF) Benchmark Test 1 Curriculum Standards: Reason about operations with real numbers. Solve exponential and logarithmic equations. Quadratic Functions 3-2: Virtual Nerd™: What is Function Notation? Curriculum Standards: Identify, evaluate, and graph linear functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Write linear functions, using function notation, to model real-world and mathematical situations. Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. Add, subtract, and multiply functions using function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. Identify, evaluate, and graph linear functions. 3-2: Ex 1: Evaluate Functions in Function Notation & Try It! Curriculum Standards: Identify, evaluate, and graph linear functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Write linear functions, using function notation, to model real-world and mathematical situations. Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. Add, subtract, and multiply functions using function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. Identify, evaluate, and graph linear functions. 2-1: Ex 1: Graph a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-2: Ex 2: Write an Equation in Point-Slope Form & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 2-2: Ex 3: Sketch the Graph of a Linear Equation in Point-Slope Form & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-2: Ex 4: Apply Linear Equations & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 2-2: Virtual Nerd™: How do you write an equation of a line in point-slope form if you have the slope and one point? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 2-2: Virtual Nerd™: How do you write an equation of a line in point-slope form if you have two points? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 2-3: Ex 2: Sketch the Graph of a Linear Equation in Standard Form & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-3: Ex 3: Relate Standard Form to Horizontal and Vertical Lines & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-3: Ex 4: Use the Standard Form of a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-3: Virtual Nerd™: How Do You Use X- and Y-Intercepts To Graph a Line In Standard Form? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 3-1: Ex 1: Recognize Domain and Range & Try It! Curriculum Standards: Determine whether a function is a relation. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determining whether a relation is a function; domain and range; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Recognize that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Extend previous knowledge of a function to apply to general behavior and features of a function. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Understand that the graph of a function labeled as ?? is the set of all ordered pairs (??,??) that satisfy the equation ??=??(??). Determine whether a function is a relation. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain. 3-2: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. Topic 3: Readiness Assessment (PDF) Curriculum Standards: Identify, evaluate, and graph linear functions. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify and describe arithmetic sequences. Identify key features of the graph of the quadratic parent function. Identify the function family when given an equation or graph. Determine whether a function is a relation. Topic 3: Readiness Assessment Curriculum Standards: Determine whether a function is a relation. Identify, evaluate, and graph linear functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. Write equations of parallel lines and perpendicular lines. Topic 3: enVision STEM Project Topic 3: enVision STEM Project (PDF) Topic 3: enVision STEM Video Topic 3: enVision STEM Masters (PDF) Key Features of a Quadratic Function Interactive Student Edition: Realize Reader: Lesson 3-1 Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Change functions to compress or stretch their graphs. Explore 3-1: Explore & Reason Understand and Apply 3-1: Ex 1: Identify a Quadratic Parent Function & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. 3-1: Ex 2: Understand the Graph of f(??) = ????² and Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Additional Example 2 with Try Another One Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. 3-1: Ex 3: Interpret Quadratic Functions from Tables & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. 3-1: Ex 4: Apply Quadratic Functions & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. 3-1: Additional Example 4 Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. 3-1: Ex 5: Compare the Rate of Change & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of the graph of the quadratic parent function. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 3-1: Concept Summary Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Change functions to compress or stretch their graphs. 3-1: Do You Understand? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Change functions to compress or stretch their graphs. 3-1: Do You Know How? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Change functions to compress or stretch their graphs. Practice and Problem Solving 3-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the function family when given an equation or graph. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Assess & Differentiate 3-1: Ex 2: Understand the Graph of f(??) = ????² and Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Virtual Nerd™: How Do You Graph the Parent Quadratic Function y=x2? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: MathXL for School: Enrichment Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Ex 3: Interpret Quadratic Functions from Tables & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. 3-1: Lesson Quiz (PDF) Curriculum Standards: Identify key features of the graph of the quadratic parent function. 3-1: Lesson Quiz Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Change functions to compress or stretch their graphs. 3-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. 3-1: Reteach to Build Understanding (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe the key features of the square root function. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. 3-1: MathXL for School: Additional Practice Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Use tables and graphs to find solutions of quadratic equations. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. 3-1: Additional Practice (PDF) Curriculum Standards: Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the function family when given an equation or graph. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. 3-1: MathXL for School: Enrichment Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Enrichment (PDF) Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. 3-1: Mathematical Literacy and Vocabulary (PDF) 3-1: Virtual Nerd™: What is a Quadratic Function? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. 3-1: Virtual Nerd™: How Do You Graph the Parent Quadratic Function y=x2? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Quadratic Functions in Vertex Form Interactive Student Edition: Realize Reader: Lesson 3-2 Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Explore 3-2: Critique & Explain Understand and Apply 3-2: Ex 1: Understand the Graph of ??(??) = ??² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Ex 2: Understand the Graph of ??(??) = (?? - h)² & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Ex 3: Understand the Graph of f(??) = ?? (?? - h)² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Ex 4: Graph Using Vertex Form & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: Ex 5: Use Vertex Form to Solve Problems & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: Additional Example 5 Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: Concept Summary Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Do You Understand? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Do You Know How? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Practice and Problem Solving 3-2: MathXL for School: Practice & Problem Solving Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Write and graph quadratic functions in standard form. Describe the key features of the square root function. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Assess & Differentiate 3-2: Ex 1: Understand the Graph of ??(??) = ??² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Ex 3: Understand the Graph of f(??) = ?? (?? - h)² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: Ex 4: Graph Using Vertex Form & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Graph quadratic functions using the vertex form. 3-2: Virtual Nerd™: What is Vertex Form of a Quadratic Equation? Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: Lesson Quiz (PDF) Curriculum Standards: Predict the behavior of polynomial functions. 3-2: Lesson Quiz Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Graph quadratic functions using the vertex form. 3-2: Reteach to Build Understanding (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: MathXL for School: Additional Practice Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Use tables and graphs to find solutions of quadratic equations. Graph quadratic functions using the vertex form. 3-2: Additional Practice (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: MathXL for School: Enrichment Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. 3-2: Enrichment (PDF) 3-2: Mathematical Literacy and Vocabulary (PDF) 3-2: Virtual Nerd™: What is Vertex Form of a Quadratic Equation? Curriculum Standards: Graph quadratic functions using the vertex form. 3-2: Virtual Nerd™: How Do You Graph a Quadratic Equation in Vertex Form? Curriculum Standards: Graph quadratic functions using the vertex form. Quadratic Functions in Standard Form Interactive Student Edition: Realize Reader: Lesson 3-3 Curriculum Standards: Graph quadratic functions using standard form. Explore 3-3: Explore & Reason Curriculum Standards: Graph quadratic functions using standard form. Understand and Apply 3-3: Ex 1: Relate c to the Graph of f(??) = ????² + ???? + ?? & Try It! Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Concept: Standard Form of a Quadratic Equation 3-3: Ex 2: Graph a Quadratic Function in Standard Form & Try It! Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Additional Example 2 Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Ex 3: Compare Properties of Quadratic Functions & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-3: Ex 4: Analyze the Structure of Different Forms & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-3: Concept Summary Curriculum Standards: Graph quadratic functions using standard form. 3-3: Do You Understand? Curriculum Standards: Graph quadratic functions using standard form. 3-3: Do You Know How? Curriculum Standards: Graph quadratic functions using standard form. Practice and Problem Solving 3-3: MathXL for School: Practice & Problem Solving Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Create and use graphs of sine and cosine functions. Assess & Differentiate 3-3: Ex 1: Relate c to the Graph of f(??) = ????² + ???? + ?? & Try It! Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Ex 3: Compare Properties of Quadratic Functions & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-3: Ex 4: Analyze the Structure of Different Forms & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-3: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Identify key features of the graph of the quadratic parent function. Graph quadratic functions using the vertex form. Graph quadratic functions using standard form. 3-3: Virtual Nerd™: What is the Standard Form of a Quadratic? Curriculum Standards: Graph quadratic functions using standard form. 3-3: Lesson Quiz (PDF) Curriculum Standards: Predict the behavior of polynomial functions. 3-3: Lesson Quiz Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. 3-3: Reteach to Build Understanding (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe the key features of the square root function. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. 3-3: MathXL for School: Additional Practice Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: Additional Practice (PDF) Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-3: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Identify key features of the graph of the quadratic parent function. Graph quadratic functions using the vertex form. Graph quadratic functions using standard form. 3-3: Enrichment (PDF) 3-3: Mathematical Literacy and Vocabulary (PDF) 3-3: Virtual Nerd™: What is the Standard Form of a Quadratic? Curriculum Standards: Graph quadratic functions using standard form. 3-3: Virtual Nerd™: How Do You Graph a Quadratic Function? Curriculum Standards: Graph quadratic functions using standard form. Modeling With Quadratic Functions Interactive Student Edition: Realize Reader: Lesson 3-4 Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Explore 3-4: Model & Discuss Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Understand and Apply 3-4: Ex 1: Use Quadratic Functions to Model Area & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Additional Example 1 with Try Another One Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Concept: Vertical Motion Model 3-4: Ex 2: Model Vertical Motion & Try It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Additional Example 2 Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 3: Assess the Fit of a Function by Analyzing Residuals & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 4: Fit a Quadratic Function to Data & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Concept Summary Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Do You Understand? Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Do You Know How? Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Practice and Problem Solving 3-4: MathXL for School: Practice & Problem Solving Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Assess & Differentiate 3-4: Ex 1: Use Quadratic Functions to Model Area & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 3: Assess the Fit of a Function by Analyzing Residuals & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 4: Fit a Quadratic Function to Data & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: MathXL for School: Enrichment Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. domain, range, and continuity; extrema; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. Graph linear and quadratic functions and show intercepts, maxima, and minima. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 3-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph quadratic functions using standard form. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between functions and other relations defined symbolically, graphically or in tabular form. Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. 3-4: Lesson Quiz (PDF) 3-4: Lesson Quiz Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph quadratic functions using standard form. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between functions and other relations defined symbolically, graphically or in tabular form. Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. 3-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: MathXL for School: Additional Practice Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Additional Practice (PDF) Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: MathXL for School: Enrichment Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. domain, range, and continuity; extrema; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. Graph linear and quadratic functions and show intercepts, maxima, and minima. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 3-4: Enrichment (PDF) Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Virtual Nerd™: How do you solve for the displacement of an object that rises and falls near Earth, given initial upward velocity, and time? Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Mathematical Modeling in 3 Acts: The Long Shot Topic 3: The Long Shot - Act 1 Video with Questions Topic 3: The Long Shot - Act 2 Content Topic 3: The Long Shot - Act 2 Questions Topic 3: The Long Shot - Act 3 Video Topic 3: The Long Shot - Act 3 Questions Linear, Exponential, and Quadratic Models Interactive Student Edition: Realize Reader: Lesson 3-5 Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Explore 3-5: Model & Discuss Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Understand and Apply 3-5: Ex 1: Determine Which Function Type Represents Data & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Additional Example 1 with Try Another One Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Ex 2: Choose a Function Type for Real-World Data & Try It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Additional Example 2 Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Ex 3: Compare Linear, Exponential, and Quadratic Growth & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-5: Concept Summary Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Do You Understand? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Do You Know How? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Practice and Problem Solving 3-5: MathXL for School: Practice & Problem Solving Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Assess & Differentiate 3-5: Ex 1: Determine Which Function Type Represents Data & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Ex 3: Compare Linear, Exponential, and Quadratic Growth & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Virtual Nerd™: How Do You Determine if a Graph Represents a Linear, Exponential, or Quadratic Function? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-5: Lesson Quiz (PDF) 3-5: Lesson Quiz Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Reteach to Build Understanding (PDF) Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: MathXL for School: Additional Practice Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Additional Practice (PDF) Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: MathXL for School: Enrichment Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Enrichment (PDF) 3-5: Mathematical Literacy and Vocabulary (PDF) 3-5: Virtual Nerd™: How Do You Determine if a Graph Represents a Linear, Exponential, or Quadratic Function? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. Topic 3: MathXL for School: Topic Review Curriculum Standards: Identify, evaluate, and graph linear functions. Graph quadratic functions using the vertex form. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Determine whether a linear, exponential, or quadratic function best models a data set. Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Identify the function family when given an equation or graph. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Topic 3: Performance Assessment Form A (PDF) Curriculum Standards: Predict the behavior of polynomial functions. Topic 3: Performance Assessment Form A Curriculum Standards: Predict the behavior of polynomial functions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Determine whether a function is a relation. Use the relationships between sides, segments, and angles of triangles to solve problems. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Topic 3: Performance Assessment Form B Topic 3: Performance Assessment Form B (PDF) 3-5: Ex 1: Determine Which Function Type Represents Data & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 3: Assess the Fit of a Function by Analyzing Residuals & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-3: Ex 2: Graph a Quadratic Function in Standard Form & Try It! Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-1: Virtual Nerd™: How Do You Graph the Parent Quadratic Function y=x2? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Ex 2: Understand the Graph of f(??) = ????² and Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-3: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-1: Ex 5: Compare the Rate of Change & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of the graph of the quadratic parent function. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 3-1: MathXL for School: Enrichment Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Additional Example 1 with Try Another One Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 1: Use Quadratic Functions to Model Area & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Additional Example 1 with Try Another One Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-2: Ex 2: Understand the Graph of ??(??) = (?? - h)² & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-5: Virtual Nerd™: How Do You Determine if a Graph Represents a Linear, Exponential, or Quadratic Function? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-5: Ex 3: Compare Linear, Exponential, and Quadratic Growth & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-3: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Identify key features of the graph of the quadratic parent function. Graph quadratic functions using the vertex form. Graph quadratic functions using standard form. 3-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph quadratic functions using standard form. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between functions and other relations defined symbolically, graphically or in tabular form. Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. 3-1: Ex 3: Interpret Quadratic Functions from Tables & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. 3-4: Ex 4: Fit a Quadratic Function to Data & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-3: Ex 3: Compare Properties of Quadratic Functions & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-4: MathXL for School: Enrichment Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. domain, range, and continuity; extrema; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. Graph linear and quadratic functions and show intercepts, maxima, and minima. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 3-3: Additional Example 2 Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-2: Ex 3: Understand the Graph of f(??) = ?? (?? - h)² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-3: Virtual Nerd™: How Do You Graph a Quadratic Function? Curriculum Standards: Graph quadratic functions using standard form. Topic 3: Assessment Form A (PDF) Curriculum Standards: Identify, evaluate, and graph linear functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 3: Assessment Form A Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Determine whether a linear, exponential, or quadratic function best models a data set. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Topic 3: Assessment Form B Topic 3: Assessment Form B (PDF) 3-5: Ex 1: Determine Which Function Type Represents Data & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 3: Assess the Fit of a Function by Analyzing Residuals & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-3: Ex 2: Graph a Quadratic Function in Standard Form & Try It! Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-1: Virtual Nerd™: How Do You Graph the Parent Quadratic Function y=x2? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Ex 2: Understand the Graph of f(??) = ????² and Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-3: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-1: Ex 5: Compare the Rate of Change & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of the graph of the quadratic parent function. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 3-1: MathXL for School: Enrichment Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-5: Additional Example 1 with Try Another One Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Ex 1: Use Quadratic Functions to Model Area & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: Additional Example 1 with Try Another One Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-2: Ex 2: Understand the Graph of ??(??) = (?? - h)² & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-5: Virtual Nerd™: How Do You Determine if a Graph Represents a Linear, Exponential, or Quadratic Function? Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-5: Ex 3: Compare Linear, Exponential, and Quadratic Growth & Try It! Curriculum Standards: Determine whether a linear, exponential, or quadratic function best models a data set. Determine whether a linear, exponential, or quadratic function best models a data set. 3-3: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Identify key features of the graph of the quadratic parent function. Graph quadratic functions using the vertex form. Graph quadratic functions using standard form. 3-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph quadratic functions using standard form. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between functions and other relations defined symbolically, graphically or in tabular form. Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. 3-1: Ex 3: Interpret Quadratic Functions from Tables & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. 3-4: Ex 4: Fit a Quadratic Function to Data & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-3: Ex 3: Compare Properties of Quadratic Functions & Try It! Curriculum Standards: Graph quadratic functions using standard form. 3-4: MathXL for School: Enrichment Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. domain, range, and continuity; extrema; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. Graph linear and quadratic functions and show intercepts, maxima, and minima. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 3-3: Additional Example 2 Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. 3-2: Ex 3: Understand the Graph of f(??) = ?? ( ( - h)² + ?? & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 3-3: Virtual Nerd™: How Do You Graph a Quadratic Function? Curriculum Standards: Graph quadratic functions using standard form. Topic 3: Assessment Form C Curriculum Standards: Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Determine whether a linear, exponential, or quadratic function best models a data set. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Solving Quadratic Equations 5-1: Ex 1: Write Radicals Using Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.) Instructional Note: Address this standard before discussing exponential functions with continuous domains. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-3: Virtual Nerd™: How Do You Solve a Formula for a Variable? Curriculum Standards: Rewrite and use literal equations to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. literal equations for a specified variable; practical problems involving equations and systems of equations. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. Solve equations involving several variables for one variable in terms of the others. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. Rewrite and use literal equations to solve problems. 1-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Rewrite and use literal equations to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. literal equations for a specified variable; practical problems involving equations and systems of equations. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. Solve equations involving several variables for one variable in terms of the others. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. Rewrite and use literal equations to solve problems. 2-3: Virtual Nerd™: How Do You Write an Equation of a Line in Standard Form from a Word Problem? Curriculum Standards: Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using standard form. 1-3: Ex 1: Rewrite Literal Equations & Try It! Curriculum Standards: Rewrite and use literal equations to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. literal equations for a specified variable; practical problems involving equations and systems of equations. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. Solve equations involving several variables for one variable in terms of the others. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. Rewrite and use literal equations to solve problems. 4-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 2-2: Ex 1: Understand Point-Slope Form of a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using point-slope form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using point-slope form. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. applying the laws of exponents to perform operations on expressions; square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.) Instructional Note: Address this standard before discussing exponential functions with continuous domains. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Rewrite expressions involving simple radicals and rational exponents in different forms. Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and Rewrite expressions involving radicals and rational exponents using the properties of exponents. Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. Understand and apply the relationship of rational exponents to integer exponents and radicals to solve problems. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. 2-5: Virtual Nerd™: How Do You Factor a Trinomial? Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 1-1: Virtual Nerd™: How Do You Solve a Multi-Step Equation with Fractions by Multiplying Away the Fraction? Curriculum Standards: Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. multistep linear equations in one variable algebraically; practical problems involving equations and systems of equations. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. 2-5: Ex 1: Understand Factoring a Trinomial & Try It! Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 3-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify, evaluate, and graph linear functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. Recognize the key features of exponential functions. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship or two input-output pairs (include reading these from a table). Instructional Note: In constructing linear functions, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.) Identify, evaluate, and graph linear functions. Recognize the key features of exponential functions. 2-2: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using point-slope form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using point-slope form. 2-1: Ex 2: Write an Equation from a Graph & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. 4-3: Ex 1: Solve a System of Equations by Adding & Try It! Curriculum Standards: Solve systems of linear equations using the elimination method. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. Solve systems of linear equations using linear combination. Solve systems of linear equations using the elimination method. Solve systems of linear equations using the elimination method. 1-1: Ex 3: Operations With Rational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. Reason about operations with real numbers. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Instructional Note: Connect to physical situations (e.g., finding the perimeter of a square of area 2). Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 2-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. domain, range, and continuity; extrema; intercepts; values of a function for elements in its domain; end behavior; Interpret parts of an expression, such as terms, factors, and coefficients. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. 2-3: Ex 1: Understand Standard Form of a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using standard form. 2-1: Virtual Nerd™: How Do You Write an Equation of a Line in Slope-Intercept Form if You Have a Graph? Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. 3-2: Ex 4: Use Linear Functions to Solve Problems & Try It! Curriculum Standards: Identify, evaluate, and graph linear functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation. Identify, evaluate, and graph linear functions. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain. 2-1: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. 4-3: MathXL for School: Enrichment Curriculum Standards: Solve systems of linear equations using the elimination method. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. Solve systems of linear equations using linear combination. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the elimination method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Solve systems of linear equations using the elimination method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 1-1: Ex 1: Solve Linear Equations & Try It! Curriculum Standards: Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. multistep linear equations in one variable algebraically; practical problems involving equations and systems of equations. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. 4-3: Virtual Nerd™: How Do You Solve a System of Equations Using the Elimination by Multiplication Method? Curriculum Standards: Solve systems of linear equations using the elimination method. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. Solve systems of linear equations using linear combination. Solve systems of linear equations using the elimination method. Solve systems of linear equations using the elimination method. 4-2: Ex 1: Solve Systems of Equations Using Substitution & Try It! Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 4-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve systems of linear equations using the elimination method. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. Solve systems of linear equations using linear combination. Solve systems of linear equations using the elimination method. Solve systems of linear equations using the elimination method. 4-2: Virtual Nerd™: What is Another Way of Solving a System of Equations Using the Substitution Method? Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Write equations of parallel lines and perpendicular lines. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. Write equations of parallel lines and perpendicular lines. 2-2: Virtual Nerd™: How do you write an equation of a line in point-slope form if you have the slope and one point? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 2-2: Ex 2: Write an Equation in Point-Slope Form & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write equations of parallel lines and perpendicular lines. 1-3: MathXL for School: Enrichment Curriculum Standards: Rewrite and use literal equations to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. literal equations for a specified variable; practical problems involving equations and systems of equations. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. Solve equations involving several variables for one variable in terms of the others. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?? = ???? to highlight resistance ??. Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. Rewrite and use literal equations to solve problems. Topic 4: Readiness Assessment (PDF) Curriculum Standards: Identify, evaluate, and graph linear functions. Write and graph linear equations using standard form. Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. Solve systems of linear equations using the elimination method. Rewrite and use literal equations to solve problems. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Reason about operations with real numbers. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Write and graph linear equations using point-slope form. Write and graph linear equations using slope-intercept form. Write equations of parallel lines and perpendicular lines. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Topic 4: Readiness Assessment Curriculum Standards: Identify, evaluate, and graph linear functions. Write and graph linear equations using standard form. Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. Solve systems of linear equations using the elimination method. Rewrite and use literal equations to solve problems. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Reason about operations with real numbers. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Write and graph linear equations using point-slope form. Write and graph linear equations using slope-intercept form. Write equations of parallel lines and perpendicular lines. Use properties of exponents to solve equations with rational exponents. Relate roots and rational exponents and use them to simplify expressions and solve equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Topic 4: enVision STEM Project Topic 4: enVision STEM Project (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 4: enVision STEM Video Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 4: enVision STEM Masters (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solving Quadratic Equations Using Graphs and Tables Interactive Student Edition: Realize Reader: Lesson 4-1 Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Explore 4-1: Explore & Reason Understand and Apply 4-1: Ex 1: Recognize Solutions of Quadratic Equations & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Additional Example 1 with Try Another One Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Ex 2: Solve Quadratic Equations Using Tables & Try It! Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. 4-1: Ex 3: Use Approximate Solutions & Try It! Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. 4-1: Additional Example 3 Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. 4-1: Concept Summary Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Do You Understand? Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Do You Know How? Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Practice and Problem Solving 4-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess & Differentiate 4-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. 4-1: Ex 3: Use Approximate Solutions & Try It! Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. 4-1: Additional Example 3 Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. 4-1: MathXL for School: Enrichment Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Lesson Quiz (PDF) Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. 4-1: Lesson Quiz Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. 4-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. 4-1: Reteach to Build Understanding (PDF) Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. 4-1: MathXL for School: Additional Practice Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. 4-1: Additional Practice (PDF) Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Identify key features of quadratic functions. 4-1: MathXL for School: Enrichment Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Enrichment (PDF) Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-1: Virtual Nerd™: How Do You Solve a Word Problem by Graphing a Quadratic Equation? Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. 4-1: Virtual Nerd™: How Do You Solve a Quadratic Equation With Two Solutions by Graphing? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solving Quadratic Equations by Factoring Interactive Student Edition: Realize Reader: Lesson 4-2 Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Explore 4-2: Model & Discuss Understand and Apply 4-2: Ex 1: Use the Zero-Product Property & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-2: Additional Example 1 with Try Another One Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-2: Ex 2: Solve by Factoring & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Ex 3: Use Factoring to Solve a Real-World Problem & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Additional Example 3 Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Ex 4: Use Factored Form to Graph a Quadratic Function & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. 4-2: Ex 5: Write the Factored Form of a Quadratic Function & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. 4-2: Concept Summary Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Do You Understand? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Do You Know How? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Practice and Problem Solving 4-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess & Differentiate 4-2: Ex 1: Use the Zero-Product Property & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-2: Additional Example 1 with Try Another One Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-2: Virtual Nerd™: What's the Zero Product Property? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-2: Ex 2: Solve by Factoring & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Virtual Nerd™: How Do You Solve a Word Problem by Factoring a Quadratic Equation? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: MathXL for School: Enrichment Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Lesson Quiz (PDF) Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Lesson Quiz Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Reteach to Build Understanding (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: MathXL for School: Additional Practice Curriculum Standards: Find the solution of a quadratic equation by factoring. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Additional Practice (PDF) Curriculum Standards: Find the solution of a quadratic equation by factoring. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: MathXL for School: Enrichment Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Enrichment (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-2: Virtual Nerd™: How Do You Solve a Word Problem by Factoring a Quadratic Equation? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-2: Virtual Nerd™: What's the Zero Product Property? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Rewriting Radical Expressions Interactive Student Edition: Realize Reader: Lesson 4-3 Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Explore 4-3: Explore & Reason Understand and Apply 4-3: Ex 1: Use Properties to Rewrite Radical Expressions & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Ex 2: Write Equivalent Radical Expressions & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Additional Example 2 with Try Another One Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Ex 3: Write Equivalent Radical Expressions With Variables & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Ex 4: Multiply Radical Expressions & Try It! Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Additional Example 4 Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Ex 5: Write a Radical Expression & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Concept Summary Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Do You Understand? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Do You Know How? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Practice and Problem Solving 4-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Assess & Differentiate 4-3: Ex 4: Multiply Radical Expressions & Try It! Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Virtual Nerd™: How Do You Multiply Two Radicals? Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: MathXL for School: Enrichment Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Lesson Quiz (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Lesson Quiz Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Reteach to Build Understanding (PDF) Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: MathXL for School: Additional Practice Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Additional Practice (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: MathXL for School: Enrichment Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Enrichment (PDF) Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Virtual Nerd™: How Do You Multiply Two Radicals? Curriculum Standards: Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-3: Virtual Nerd™: What is the Product Property of Square Roots? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Solving Quadratic Equations Using Square Roots Interactive Student Edition: Realize Reader: Lesson 4-4 Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Explore 4-4: Explore & Reason Understand and Apply 4-4: Ex 1: Solve Equations of the Form ??² = ?? & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Additional Example 1 with Try Another One Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Additional Example 2 Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Ex 3: Solve Equations of the Form ????² + ?? = ?? & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Ex 4: Determine a Reasonable Solution & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: Concept Summary Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: Do You Understand? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: Do You Know How? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Practice and Problem Solving 4-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Assess & Differentiate 4-4: Ex 4: Determine a Reasonable Solution & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Lesson Quiz (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: Lesson Quiz Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 4-4: Reteach to Build Understanding (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: MathXL for School: Additional Practice Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: Additional Practice (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Enrichment (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Virtual Nerd™: What is the Square Root Property? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Mathematical Modeling in 3 Acts: Unwrapping Change Topic 4: Unwrapping Change - Act 1 Video With Questions Topic 4: Unwrapping Change - Act 2 Content Topic 4: Unwrapping Change - Act 2 Questions Topic 4: Unwrapping Change - Act 3 Video Topic 4: Unwrapping Change - Act 3 Questions Solving Systems of Linear and Quadratic Equations Interactive Student Edition: Realize Reader: Lesson 4-5 Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Explore 4-5: Model & Discuss Understand and Apply 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Additional Example 1 Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Ex 2: Solve a Linear-Quadratic Equation by Graphing & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Additional Example 2 with Try Another One Curriculum Standards: Solve a system with linear and quadratic equations. 4-5: Ex 3: Solve Systems of Equations Using Elimination & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Ex 4: Solve Systems Using Substitution & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Concept Summary Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Do You Understand? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Do You Know How? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Practice and Problem Solving 4-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Assess & Differentiate 4-5: Ex 2: Solve a Linear-Quadratic Equation by Graphing & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Virtual Nerd™: How Do You Solve a System of Equations by Graphing if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 4-5: MathXL for School: Enrichment Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ² ² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Lesson Quiz (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Lesson Quiz Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 4-5: Reteach to Build Understanding (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: MathXL for School: Additional Practice Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Additional Practice (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: MathXL for School: Enrichment Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Enrichment (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. 4-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 4-5: Virtual Nerd™: How Do You Solve a System of Equations by Graphing if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Topic 4: MathXL for School: Topic Review Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Find the solution of a quadratic equation by factoring. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Identify key features of quadratic functions. Use properties of exponents to solve equations with rational exponents. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 4: Performance Assessment Form A (PDF) Topic 4: Performance Assessment Form A Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Use graphs and tables to approximate solutions to algebraic equations and inequalities. Solve exponential and logarithmic equations. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Use the relationships between sides, segments, and angles of triangles to solve problems. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Topic 4: Performance Assessment Form B Topic 4: Performance Assessment Form B (PDF) 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Topic 4: Assessment Form A (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 4: Assessment Form A Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 4: Assessment Form B Topic 4: Assessment Form B (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers f and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Topic 4: Assessment Form C Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 1-2: Virtual Nerd™: What are Rational Exponents? Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 3-4: Ex 3: Assess the Fit of a Function by Analyzing Residuals & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-1: Virtual Nerd™: How Do You Graph the Parent Quadratic Function y=x2? Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??((?) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Ex 2: Analyze Horizontal Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ),(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??((?+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 3-1: Ex 2: Understand the Graph of f(??) = ????² and Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 3-1: Ex 5: Compare the Rate of Change & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Identify key features of the graph of the quadratic parent function. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Note: Emphasize the selection of a model function based on behavior of data and context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values. Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. 3-1: MathXL for School: Enrichment Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 4-2: Ex 1: Use the Zero-Product Property & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-5: Virtual Nerd™: How Do You Solve a System of Equations by Graphing if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 1-2: Ex 4: Use the Power of a Product Property to Solve Equations With Rational Exponents & Try It! Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 6-6: Virtual Nerd™: What Does the Constant 'h' Do in the Function ??(??)=v(??-h)? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 4-2: Virtual Nerd™: How Do You Solve a Word Problem by Factoring a Quadratic Equation? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 3-4: Ex 1: Use Quadratic Functions to Model Area & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 4-2: Additional Example 1 with Try Another One Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 3-4: Additional Example 1 with Try Another One Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 4-5: Ex 2: Solve a Linear-Quadratic Equation by Graphing & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 1-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-2: Ex 2: Solve by Factoring & Try It! Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 6-6: Ex 3: Combine Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??((???), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??((?)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Virtual Nerd™: What Does the Constant '??' Do in the Function ??(??)=v(??)+??? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??((? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??++?), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 4-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: MathXL for School: Enrichment Curriculum Standards: Solve a system with linear and quadratic equations. Solve a system with linear and quadratic equations. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 6-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 3-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph quadratic functions using standard form. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.) Instructional Note: Focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Distinguish between functions and other relations defined symbolically, graphically or in tabular form. Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. 6-6: MathXL for School: Enrichment Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 3-4: Ex 4: Fit a Quadratic Function to Data & Try It! Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 3-4: MathXL for School: Enrichment Curriculum Standards: Use quadratic functions to model real-world situations. Write and graph quadratic functions in standard form. Graph quadratic functions using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. domain, range, and continuity; extrema; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. Graph linear and quadratic functions and show intercepts, maxima, and minima. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 1-2: MathXL for School: Enrichment Curriculum Standards: Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. 4-2: Virtual Nerd™: What's the Zero Product Property? Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Factor a quadratic expression to reveal the zeros of the function it defines. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 4-2: MathXL for School: Enrichment Curriculum Standards: Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Solve problems with complex numbers. Solve quadratic equations using the Quadratic Formula. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 1-1: Ex 3: Identify Positive or Negative Intervals & Try It! Curriculum Standards: Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. 1-1: MathXL for School: Enrichment Curriculum Standards: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. Benchmark Test 2 (PDF) Benchmark Test 2 Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Write and graph quadratic functions in standard form. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the solution of a quadratic equation by factoring. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Use the unit circle to evaluate the trigonometric functions of any angle. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Identify the function family when given an equation or graph. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Quadratic Equations and Complex Numbers 2-1: Ex 4: Interpret Slope and y-Intercept & Try It! Curriculum Standards: Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 2-2: Ex 1: Understand Point-Slope Form of a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using point-slope form. 2-2: Virtual Nerd™: How Do You Multiply Binomials Using the Distributive Property? Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-7: Ex 3: Factor a Difference of Two Squares & Try It! Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 1-1: Ex 4: Operations With Rational and Irrational Numbers & Try It! Curriculum Standards: Reason about operations with real numbers. 1-2: Ex 1: Solving Equations With a Variable on Both Sides & Try It! Curriculum Standards: Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Topic 5: Readiness Assessment (PDF) Topic 5: Readiness Assessment Curriculum Standards: Write and graph linear equations using standard form. Reason about operations with real numbers. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Find the solution of a quadratic equation by factoring. Find the zeros of quadratic functions. Understand and apply the geometric properties of a parabola. Understand and apply the geometric properties of a hyperbola. Write and graph linear equations using point-slope form. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Topic 5: enVision STEM Project Topic 5: enVision STEM Project (PDF) Topic 5: enVision STEM Video Topic 5: enVision STEM Masters (PDF) Complex Numbers and Operations Interactive Student Edition: Realize Reader: Lesson 5-1 Explore 5-1: Explore & Reason Curriculum Standards: Solve problems with complex numbers. Understand and Apply 5-1: Ex 1: Solve a Quadratic Equation Using Square Roots & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Concept: Complex Numbers 5-1: Ex 2: Add and Subtract Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. 5-1: Additional Example 2 with Try Another One Curriculum Standards: Solve problems with complex numbers. 5-1: Ex 3: Multiply Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: Concept: Complex Conjugates 5-1: Ex 4: Simplify a Quotient With Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; 5-1: Additional Example 4 5-1: Ex 5: Factor a Sum of Squares & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??² + 4 as (?? + 2??)(?? – 2??). 5-1: Ex 6: Solve a Quadratic Equation With Complex Solutions & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-1: Concept Summary 5-1: Do You Understand? Curriculum Standards: Solve problems with complex numbers. 5-1: Do You Know How? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Practice and Problem Solving 5-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Solve problems with complex numbers. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess & Differentiate 5-1: Ex 4: Simplify a Quotient With Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; 5-1: Ex 5: Factor a Sum of Squares & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??² + 4 as (?? + 2??)(?? – 2??). 5-1: Ex 2: Add and Subtract Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. 5-1: Additional Example 2 with Try Another One Curriculum Standards: Solve problems with complex numbers. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: Ex 1: Solve a Quadratic Equation Using Square Roots & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Ex 3: Multiply Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: MathXL for School: Enrichment Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: Lesson Quiz (PDF) 5-1: Lesson Quiz Curriculum Standards: Solve problems with complex numbers. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: Reteach to Build Understanding (PDF) Curriculum Standards: Solve problems with complex numbers. 5-1: MathXL for School: Additional Practice Curriculum Standards: Solve problems with complex numbers. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-1: Additional Practice (PDF) 5-1: MathXL for School: Enrichment Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: Enrichment (PDF) Curriculum Standards: Solve problems with complex numbers. 5-1: Mathematical Literacy and Vocabulary (PDF) 5-1: Virtual Nerd™: What is the Difference Between Imaginary and Complex Numbers? Curriculum Standards: Solve problems with complex numbers. 5-1: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Mathematical Modeling in 3 Acts: Swift Kick Topic 5: Swift Kick - Act 1 Video with Questions Topic 5: Swift Kick - Act 2 Content Topic 5: Swift Kick - Act 2 Questions Topic 5: Swift Kick - Act 3 Video Topic 5: Swift Kick - Act 3 Questions Completing the Square Interactive Student Edition: Realize Reader: Lesson 5-2 Explore 5-2: Critique & Explain Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Understand and Apply 5-2: Ex 1: Use Square Roots to Solve Quadratic Equations & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-2: Ex 2: Understand the Process of Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (²)² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 3: Solve a Quadratic Equation by Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 4: Complete the Square to Solve a Real-World Problem & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Determine the maximum or minimum value of a quadratic function by completing the square. 5-2: Additional Example 4 Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. 5-2: Ex 5: Write a Quadratic Equation in Vertex Form & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. 5-2: Additional Example 5 with Try Another One Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. 5-2: Concept Summary Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 5-2: Do You Understand? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-2: Do You Know How? Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Practice and Problem Solving 5-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess & Differentiate 5-2: Ex 4: Complete the Square to Solve a Real-World Problem & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Determine the maximum or minimum value of a quadratic function by completing the square. 5-2: Virtual Nerd™: How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 2: Understand the Process of Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 3: Solve a Quadratic Equation by Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: MathXL for School: Enrichment Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Lesson Quiz (PDF) 5-2: Lesson Quiz Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 5-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Reteach to Build Understanding (PDF) Curriculum Standards: Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 5-2: MathXL for School: Additional Practice Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-2: Additional Practice (PDF) 5-2: MathXL for School: Enrichment Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Enrichment (PDF) 5-2: Mathematical Literacy and Vocabulary (PDF) 5-2: Virtual Nerd™: How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Virtual Nerd™: How Do You Convert a Quadratic from Standard Form to Vertex Form by Completing the Square? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. The Quadratic Formula Interactive Student Edition: Realize Reader: Lesson 5-3 Explore 5-3: Explore & Reason Understand and Apply 5-3: Ex 1: Solve Quadratic Equations & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Additional Example 1 with Try Another One Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Ex 2: Choose a Solution Method & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Ex 3: Identify the Number of Real-Number Solutions & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Additional Example 3 Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Ex 4: Interpret the Discriminant & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Ex 5: Use the Discriminant to Find a Particular Equation & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Enrichment Example Relate the and the Number of Zeros Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Concept Summary Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Do You Understand? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Do You Know How? Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Practice and Problem Solving 5-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Use tables and graphs to find solutions of quadratic equations. Identify key features of quadratic functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Assess & Differentiate 5-3: Ex 1: Solve Quadratic Equations & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. 5-3: MathXL for School: Enrichment Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Ex 4: Interpret the Discriminant & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Find the Discriminant of a Quadratic Equation With 2 Complex Solutions? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Lesson Quiz (PDF) 5-3: Lesson Quiz Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for mb² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. 5-3: Reteach to Build Understanding (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: MathXL for School: Additional Practice Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Additional Practice (PDF) 5-3: MathXL for School: Enrichment Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Enrichment (PDF) 5-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. 5-3: Virtual Nerd™: How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Find the Discriminant of a Quadratic Equation With 2 Complex Solutions? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Linear Quadratic Systems Interactive Student Edition: Realize Reader: Lesson 5-4 Explore 5-4: Explore & Reason Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Understand and Apply 5-4: Ex 1: Determine the Number of Solutions & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Ex 2: Solve a Linear-Quadratic System Using Substitution & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Ex 3: Applying a Linear-Quadratic System & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Additional Example 3 Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Ex 4: Solve a Linear-Quadratic System of Inequalities & Try It! Curriculum Standards: Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Solve linear-quadratic systems. 5-4: Ex 5: Using a System to Solve an Equation & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Additional Example 5 with Try Another One Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Concept Summary Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Do You Understand? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Do You Know How? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Practice and Problem Solving 5-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Assess & Differentiate 5-4: Ex 4: Solve a Linear-Quadratic System of Inequalities & Try It! Curriculum Standards: Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Solve linear-quadratic systems. 5-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. 5-4: Ex 2: Solve a Linear-Quadratic System Using Substitution & Try It! Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Virtual Nerd™: How Do You Solve a Linear-Quadratic System Using Substitution? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: MathXL for School: Enrichment Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Lesson Quiz (PDF) 5-4: Lesson Quiz Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. 5-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. 5-4: Reteach to Build Understanding (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: MathXL for School: Additional Practice Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. 5-4: Additional Practice (PDF) 5-4: MathXL for School: Enrichment Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Enrichment (PDF) 5-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. 5-4: Virtual Nerd™: How Do You Solve a Linear-Quadratic System Using Substitution? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Polynomial Identities Interactive Student Edition: Realize Reader: Lesson 5-5 Explore 5-5: Explore & Reason Understand and Apply 5-5: Concept: Polynomial Identities 5-5: Ex 1: Prove a Polynomial Identity & Try It! Curriculum Standards: Prove and use polynomial identities. 5-5: Additional Example 1 with Try Another One Curriculum Standards: Prove and use polynomial identities. 5-5: Ex 2: Use Polynomial Identities to Multiply & Try It! Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: Additional Example 2 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: Ex 3: Use Polynomial Identities to Factor and Simplify & Try It! Curriculum Standards: Prove and use polynomial identities. 5-5: Ex 4: Expand a Power of a Binomial & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Concept: Binomial Theorem 5-5: Ex 5: Apply the Binomial Theorem & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Enrichment Example Generate Pythagorean Triples Curriculum Standards: Prove and use polynomial identities. 5-5: Concept Summary 5-5: Do You Understand? Curriculum Standards: Prove and use polynomial identities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Do You Know How? Curriculum Standards: Prove and use polynomial identities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Practice and Problem Solving 5-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Prove and use polynomial identities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Assess & Differentiate 5-5: Ex 4: Expand a Power of a Binomial & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: MathXL for School: Enrichment Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Ex 1: Prove a Polynomial Identity & Try It! Curriculum Standards: Prove and use polynomial identities. 5-5: Additional Example 2 Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: Ex 3: Use Polynomial Identities to Factor and Simplify & Try It! Curriculum Standards: Prove and use polynomial identities. 5-5: Ex 5: Apply the Binomial Theorem & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Virtual Nerd™: How Do You Expand a Power of a Binomial Sum Using the Binomial Theorem? Curriculum Standards: Prove and use polynomial identities. 2-2: Ex 2: Multiply Polynomials & Try It! Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: Virtual Nerd™: How Do You Solve a Word Problem by Subtracting and Multiplying Polynomials? Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 2-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Multiply two polynomials. Use patterns to multiply binomials. Add, subtract, and multiply polynomials. Prove and use polynomial identities. factor polynomials completely in one or two variables. extrema; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. Add, subtract, multiply, divide, and simplify polynomial and rational expressions. Combine like terms to simplify polynomials. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. 5-5: Lesson Quiz (PDF) 5-5: Lesson Quiz Curriculum Standards: Prove and use polynomial identities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: Reteach to Build Understanding (PDF) Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. 5-5: MathXL for School: Additional Practice Curriculum Standards: Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. 5-5: Additional Practice (PDF) 5-5: MathXL for School: Enrichment Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-5: Enrichment (PDF) 5-5: Mathematical Literacy and Vocabulary (PDF) 5-5: Virtual Nerd™: How Do You Expand a Power of a Binomial Sum Using the Binomial Theorem? Curriculum Standards: Prove and use polynomial identities. 5-5: Virtual Nerd™: What is the Formula for Factoring the Sum of Cubes? Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Extension 5-5a: The Fundamental Theorem of Algebra Interactive Student Edition: Realize Reader: Lesson 5-5a 5-5a: Ex 1: Find the Number of Complex Roots of an Equation & Try-It 5-5a: Concept: The Fundamental Theorem of Algebra 5-5a: Ex 2: Prove the Fundamental Theorem of Algebra for Quadratics 5-5a: Do You Understand 5-5a: Do You Understand5-5a: Do You Understand 5-5a: Do You Know How 5-5a: Do You Know How5-5a: Do You Know How 5-5a: Concept Summary 5-5a: MathXL for School: Practice & Problem Solving Topic 5: MathXL for School: Topic Review Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. Solve problems with complex numbers. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Write and graph quadratic functions in standard form. Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Identify the function family when given an equation or graph. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Topic 5: Performance Assessment Form A (PDF) Topic 5: Performance Assessment Form A Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Identify key features of the graph of the quadratic parent function. Identify key features of quadratic functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Use the relationships between sides, segments, and angles of triangles to solve problems. Use tables and graphs to find solutions of quadratic equations. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Topic 5: Performance Assessment Form B Topic 5: Performance Assessment Form B (PDF) 5-2: MathXL for School: Enrichment Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-3: Ex 4: Interpret the Discriminant & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for mb² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 5-1: Ex 4: Simplify a Quotient With Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; 5-5: Ex 4: Expand a Power of a Binomial & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-3: Virtual Nerd™: How Do You Find the Discriminant of a Quadratic Equation With 2 Complex Solutions? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 5-5: Ex 3: Use Polynomial Identities to Factor and Simplify & Try It! Curriculum Standards: Prove and use polynomial identities. 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-2: Ex 2: Understand the Process of Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. 5-3: Ex 1: Solve Quadratic Equations & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 5-1: Ex 5: Factor a Sum of Squares & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??² + 4 as (?? + 2??)(?? – 2??). 5-5: MathXL for School: Enrichment Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Ex 3: Multiply Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: MathXL for School: Enrichment Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-3: MathXL for School: Enrichment Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-2: Ex 3: Solve a Quadratic Equation by Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 5-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 4: Complete the Square to Solve a Real-World Problem & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Determine the maximum or minimum value of a quadratic function by completing the square. 5-2: Virtual Nerd™: How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Topic 5: Assessment Form A (PDF) Topic 5: Assessment Form A Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve problems with complex numbers. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Prove and use polynomial identities. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Topic 5: Assessment Form B Topic 5: Assessment Form B (PDF) 5-2: MathXL for School: Enrichment Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-3: Ex 4: Interpret the Discriminant & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. square roots of whole numbers and monomial algebraic expressions; numerical expressions containing square or cube roots. quadratic equations in one variable algebraically; practical problems involving equations and systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. 5-1: Ex 4: Simplify a Quotient With Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; 5-5: Ex 4: Expand a Power of a Binomial & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 5-3: Virtual Nerd™: How Do You Find the Discriminant of a Quadratic Equation With 2 Complex Solutions? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Ex 1: Linear-Quadratic Systems of Equations Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 5-5: Ex 3: Use Polynomial Identities to Factor and Simplify & Try It! Curriculum Standards: Prove and use polynomial identities. 4-4: MathXL for School: Enrichment Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-2: Ex 2: Understand the Process of Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and a. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. 5-3: Ex 1: Solve Quadratic Equations & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Create and solve linear equations with one variable. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Create and solve linear equations with one variable. Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 5-1: Ex 5: Factor a Sum of Squares & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??² + 4 as (?? + 2??)(?? – 2??). 5-5: MathXL for School: Enrichment Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 4-4: Ex 2: Solve Equations of the Form ??x² = ?? & Try It! Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Ex 3: Multiply Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-1: MathXL for School: Enrichment Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplify, add, subtract, multiply, and divide complex numbers. Use the relation ??² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 5-3: MathXL for School: Enrichment Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-2: Ex 3: Solve a Quadratic Equation by Completing the Square & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 4-4: Virtual Nerd™: How Do You Use the Square Root Method to Solve a Quadratic Equation with Imaginary Solutions if a?1? Curriculum Standards: Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by taking square roots. Solve problems with complex numbers. Solve quadratic equations by completing the square. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 4-5: Virtual Nerd™: How Do You Solve a System of Equations Using Elimination if One Equation is a Quadratic? Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. quadratic equations over the set of complex numbers; The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?? = –3?? and the circle ??² + ??² = 3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) 5-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. 5-2: Ex 4: Complete the Square to Solve a Real-World Problem & Try It! Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Determine the maximum or minimum value of a quadratic function by completing the square. 5-2: Virtual Nerd™: How Do You Solve a Quadratic Equation with Complex Solutions by Completing the Square? Curriculum Standards: Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). Use the method of completing the square to transform any quadratic equation in ?? into an equation of the form (?? – ??)² = ?? that has the same solutions. Derive the quadratic formula from this form. Use the structure of an expression to identify ways to rewrite it. Example: For example, see ??4 – ??4 as (??²)² – (??²)², thus recognizing it as a difference of squares that can be factored as (??² – ??²)(??² + ??²). Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. Topic 5: Assessment Form C Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Solve problems with complex numbers. Use completing the square to solve quadratic equations. Solve quadratic equations by completing the square. Prove and use polynomial identities. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. Working With Functions 1-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and solve absolute-value equations and inequalities represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. multistep linear equations in one variable algebraically; practical problems involving equations and systems of equations. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Solve absolute value equations and interpret the solutions in the original context. Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Write and solve absolute-value equations and inequalities Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 2-3: Virtual Nerd™: How Do You Write an Equation of a Line in Standard Form from a Word Problem? Curriculum Standards: Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using standard form. 1-6: Ex 2: Apply an Absolute Value Equation & Try It! Curriculum Standards: Write and solve absolute-value equations and inequalities represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. multistep linear equations in one variable algebraically; practical problems involving equations and systems of equations. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Solve absolute value equations and interpret the solutions in the original context. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Write and solve absolute-value equations and inequalities Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 3-1: Ex 1: Recognize Domain and Range & Try It! Curriculum Standards: Determine whether a function is a relation. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determining whether a relation is a function; domain and range; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Recognize that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ?? is a function and ?? is an element of its domain, then ??(??) denotes the output of ?? corresponding to the input ??. The graph of ?? is the graph of the equation ?? = ??(??). Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain Extend previous knowledge of a function to apply to general behavior and features of a function. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. Represent a function using function notation and explain that ??(??) denotes the output of function ?? that corresponds to the input ??. Understand that the graph of a function labeled as ?? is the set of all ordered pairs (??,??) that satisfy the equation ??=??(??). Determine whether a function is a relation. Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain. 1-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and solve compound inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Write and solve compound inequalities. Write and solve compound inequalities. 4-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 2-3: Virtual Nerd™: How Do You Use X- and Y-Intercepts To Graph a Line In Standard Form? Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 1-5: MathXL for School: Enrichment Curriculum Standards: Write and solve compound inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Write and solve compound inequalities. Write and solve compound inequalities. 2-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Graph linear and quadratic functions and show intercepts, maxima, and minima. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 2-1: Ex 4: Interpret Slope and y-Intercept & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Use slope to solve problems about parallel and perpendicular lines. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Instructional Note: Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. Interpret parts of an expression, such as terms, factors, and coefficients. Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 2-4: Ex 2: Understand the Slopes of Perpendicular Lines & Try It! Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 2-4: Virtual Nerd™: How Do You Know if Two Lines Are Perpendicular? Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 2-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Use slope to solve problems about parallel and perpendicular lines. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Instructional Note: Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. Interpret parts of an expression, such as terms, factors, and coefficients. Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. Graph linear and quadratic functions and show intercepts, maxima, and minima. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 2-4: MathXL for School: Enrichment Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 1-5: Ex 1: Understand Compound Inequalities & Try It! Curriculum Standards: Write and solve compound inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Write and solve compound inequalities. Write and solve compound inequalities. 2-3: Ex 1: Understand Standard Form of a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write and graph linear equations using standard form. 3-3: MathXL for School: Enrichment Curriculum Standards: Transform linear equations represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and domain and range; intercepts; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on quadratic functions, and consider including absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) Transform linear equations 1-6: Ex 1: Understand Absolute Value Equations & Try It! Curriculum Standards: Write and solve absolute-value equations and inequalities represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. multistep linear equations in one variable algebraically; practical problems involving equations and systems of equations. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Solve absolute value equations and interpret the solutions in the original context. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Write and solve absolute-value equations and inequalities Create equations and inequalities in one variable and use them to solve problems. Instructional Note: Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) 3-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Transform linear equations represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and domain and range; intercepts; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on quadratic functions, and consider including absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) Transform linear equations 4-3: MathXL for School: Enrichment Curriculum Standards: Solve systems of linear equations using the elimination method. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. Solve systems of linear equations using linear combination. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the elimination method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Solve systems of linear equations using the elimination method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 3-2: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 4-2: Ex 1: Solve Systems of Equations Using Substitution & Try It! Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 3-3: Virtual Nerd™: Transforming Linear Functions Curriculum Standards: Transform linear equations represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and domain and range; intercepts; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on quadratic functions, and consider including absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) Transform linear equations 2-3: MathXL for School: Enrichment Curriculum Standards: Write and graph linear equations using slope-intercept form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Use slope to solve problems about parallel and perpendicular lines. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Instructional Note: Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. Interpret parts of an expression, such as terms, factors, and coefficients. Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. Interpret parts of an expression, such as terms, factors, and coefficients. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. Write and graph linear equations using slope-intercept form. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. 4-2: Virtual Nerd™: What is Another Way of Solving a System of Equations Using the Substitution Method? Curriculum Standards: Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. systems of two linear equations in two variables algebraically and graphically; and practical problems involving equations and systems of equations. graph linear equations in two variables. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve systems of linear equations using the substitution method. Solve systems of linear equations using the substitution method. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 3-3: Ex 2: Horizontal Translations of Linear Functions & Try It! Curriculum Standards: Transform linear equations represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and domain and range; intercepts; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on quadratic functions, and consider including absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) Transform linear equations 2-1: Ex 1: Graph a Linear Equation & Try It! Curriculum Standards: Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and graph linear equations in two variables. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph linear and quadratic functions and show intercepts, maxima, and minima. Represent and solve problems in various contexts using linear and quadratic functions. Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Write and graph linear equations using standard form. 1-5: Virtual Nerd™: How Do You Solve an AND Compound Inequality and Graph It On a Number Line? Curriculum Standards: Write and solve compound inequalities. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. solve multistep linear inequalities in one variable algebraically and represent the solution graphically; solve practical problems involving inequalities; and Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Note: Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5? = 125 or 2? = 1 /16. Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Write and solve compound inequalities. Write and solve compound inequalities. Topic 6: Readiness Assessment (PDF) Curriculum Standards: Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Solve and graph inequalities. Use the relationships between sides, segments, and angles of triangles to solve problems. Use a variety of tools to solve systems of linear equations and inequalities. Graph solutions to linear inequalities in two variables. Graph and solve a system of linear inequalities. Solve linear-quadratic systems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Topic 6: Readiness Assessment Curriculum Standards: Write and graph linear equations using standard form. Transform linear equations Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Write and solve compound inequalities. Use graphs to find approximate solutions to systems of equations. Solve systems of linear equations using the substitution method. Use a variety of tools to solve systems of linear equations and inequalities. Write and graph linear equations using slope-intercept form. Write and graph linear equations using point-slope form. Determine whether a function is a relation. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Topic 6: enVision STEM Project Topic 6: enVision STEM Project (PDF) Topic 6: enVision STEM Video Topic 6: enVision STEM Masters (PDF) The Absolute Value Function Interactive Student Edition: Realize Reader: Lesson 6-1 Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and analyze transformations of the absolute value function. Explore 6-1: Explore & Reason Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Understand and Apply 6-1: Ex 1: Graph the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: Ex 2: Transform the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. 6-1: Additional Example 2 with Try Another One Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. 6-1: Ex 3: Interpret the Graph of a Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: Additional Example 3 Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: Ex 4: Determine Rate of Change & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on linear functions and exponential functions whose domain is a subset of the integers. The Unit on Quadratic Functions and Modeling in this course and the Algebra II course address other types of functions. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.) 6-1: Concept Summary Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and analyze transformations of the absolute value function. 6-1: Do You Understand? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and analyze transformations of the absolute value function. 6-1: Do You Know How? Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and analyze transformations of the absolute value function. Practice and Problem Solving 6-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. Assess & Differentiate 6-1: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) 6-1: Ex 4: Determine Rate of Change & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. square roots of whole numbers and monomial algebraic expressions; cube roots of integers; and numerical expressions containing square or cube roots. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on linear functions and exponential functions whose domain is a subset of the integers. The Unit on Quadratic Functions and Modeling in this course and the Algebra II course address other types of functions. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Make qualitative statements about the rate of change of a function, based on its graph or table of values Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.) 6-1: Lesson Quiz (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify the function family when given an equation or graph. 6-1: Lesson Quiz Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify the function family when given an equation or graph. 6-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: Reteach to Build Understanding (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: MathXL for School: Additional Practice Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. 6-1: Additional Practice (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. 6-1: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) 6-1: Enrichment (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Identify the function family when given an equation or graph. 6-1: Mathematical Literacy and Vocabulary (PDF) 6-1: Virtual Nerd™: What is an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-1: Virtual Nerd™: How Do You Graph an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Mathematical Modeling in 3 Acts:The Mad Runner Topic 6: The Mad Runner - Act 1 Video with Questions Topic 6: The Mad Runner - Act 2 Content Topic 6: The Mad Runner - Act 2 Questions Topic 6: The Mad Runner - Act 3 Video Topic 6: The Mad Runner - Act 3 Questions Piecewise-Defined Functions Interactive Student Edition: Realize Reader: Lesson 6-2 Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Explore 6-2: Explore & Reason Understand and Apply 6-2: Ex 1: Understand Piecewise-Defined Functions & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. 6-2: Additional Example 1 with Try Another One Curriculum Standards: Graph and apply piecewise-defined functions. 6-2: Ex 2: Graph a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Ex 3: Analyze the Graph of a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: Ex 4: Apply a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-2: Additional Example 4 Curriculum Standards: Graph and apply piecewise-defined functions. 6-2: Concept Summary Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Do You Understand? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Do You Know How? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Practice and Problem Solving 6-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Assess & Differentiate 6-2: Ex 1: Understand Piecewise-Defined Functions & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. 6-2: Virtual Nerd™: What is a Piecewise Linear Function? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-2: Ex 2: Graph a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: MathXL for School: Enrichment Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: Lesson Quiz (PDF) Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Lesson Quiz Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-2: Reteach to Build Understanding (PDF) Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: MathXL for School: Additional Practice Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Additional Practice (PDF) Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: MathXL for School: Enrichment Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: Enrichment (PDF) Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. 6-2: Mathematical Literacy and Vocabulary (PDF) 6-2: Virtual Nerd™: What is a Piecewise Linear Function? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Step Functions Interactive Student Edition: Realize Reader: Lesson 6-3 Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Explore 6-3: Critique & Explain Curriculum Standards: Graph and apply step functions. Understand and Apply 6-3: Ex 1: Understand Step Functions & Try It! Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Additional Example 1 with Try Another One Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Ex 2: Use a Step Function to Represent a Real-World Situation & Try It! Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Additional Example 2 Curriculum Standards: Graph and apply step functions. 6-3: Ex 3: Use a Step Function to Solve Problems & Try It! Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Concept Summary Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Do You Understand? Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Do You Know How? Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Practice and Problem Solving 6-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Assess & Differentiate 6-3: Ex 2: Use a Step Function to Represent a Real-World Situation & Try It! Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Virtual Nerd™: What is a Step Function? Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: MathXL for School: Enrichment Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Ex 1: Understand Step Functions & Try It! Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Virtual Nerd™: How Do You Graph a Real-World Example of a Step Function? Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Lesson Quiz (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Lesson Quiz Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Reteach to Build Understanding (PDF) Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: MathXL for School: Additional Practice Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Additional Practice (PDF) Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: MathXL for School: Enrichment Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: Enrichment (PDF) Curriculum Standards: Graph and apply step functions. 6-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Graph and apply step functions. 6-3: Virtual Nerd™: How Do You Graph a Real-World Example of a Step Function? Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. 6-3: Virtual Nerd™: What is a Step Function? Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Transformations of Piecewise-Defined Functions Interactive Student Edition: Realize Reader: Lesson 6-4 Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Explore 6-4: Model & Discuss Curriculum Standards: Graph and analyze transformations of the absolute value function. Understand and Apply 6-4: Ex 1: Translate Step Functions & Try It! Curriculum Standards: Graph and analyze transformations of the absolute value function. 6-4: Ex 2: Vertical Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Additional Example 2 with Try Another One Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Ex 3: Horizontal Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Ex 4: Understand Vertical and Horizontal Translations & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Additional Example 4 Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Ex 5: Understand Vertical Stretches and Compressions & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Ex 6: Understand Transformations of the Absolute Value Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Concept Summary Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. 6-4: Do You Understand? Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. 6-4: Do You Know How? Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Practice and Problem Solving 6-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Assess & Differentiate 6-4: Ex 2: Vertical Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-1: Virtual Nerd™: What is an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Lesson Quiz (PDF) Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. 6-4: Lesson Quiz Curriculum Standards: Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Reteach to Build Understanding (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: MathXL for School: Additional Practice Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Additional Practice (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Enrichment (PDF) Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. 6-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Graph and analyze transformations of the absolute value function. 6-4: Virtual Nerd™: How Do You Write an Equation for a Translation of an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Virtual Nerd™: What Does the Constant 'a' do in y = a|x|? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyzing Functions Graphically Interactive Student Edition: Realize Reader: Lesson 6-5 Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Explore 6-5: Model & Discuss Understand and Apply 6-5: Ex 1: Analyze Domain and Range & Try It! Curriculum Standards: Identify the function family when given an equation or graph. Identify the function family when given an equation or graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) Note: Emphasize the selection of a model function based on behavior of data and context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. Find the domain of a function defined symbolically, graphically or in a real-world context. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. 6-5: Additional Example 1 with Try Another One Curriculum Standards: Identify the function family when given an equation or graph. Identify the function family when given an equation or graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) Note: Emphasize the selection of a model function based on behavior of data and context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. Find the domain of a function defined symbolically, graphically or in a real-world context. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. 6-5: Ex 2: Analyze Maximum and Minimum Values & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: Additional Example 2 Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. 6-5: Ex 3: Understand Axes of Symmetry & Try It! Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. 6-5: Ex 4: Analyze End Behaviors of Graphs & Try It! Curriculum Standards: Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; intercepts; end behavior; vertical and horizontal asymptotes; For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: Concept Summary Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. 6-5: Do You Understand? Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. 6-5: Do You Know How? Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Practice and Problem Solving 6-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Identify the key features of the cube root function. Determine whether a function is a relation. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe the key features of the square root function. Graph and transform radical functions. Find the zeros of quadratic functions. Predict the behavior of polynomial functions. Identify different types of symmetry in two-dimensional figures. Analyze functions that include absolute value expressions. investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle). Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 6-5: Ex 4: Analyze End Behaviors of Graphs & Try It! Curriculum Standards: Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; intercepts; end behavior; vertical and horizontal asymptotes; For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. end behavior; vertical and horizontal asymptotes; Analyze functions that include absolute value expressions. 6-5: Ex 2: Analyze Maximum and Minimum Values & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Analyze functions that include absolute value expressions. 6-5: Ex 3: Understand Axes of Symmetry & Try It! Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. 6-5: Virtual Nerd™: What is the Axis of Symmetry of a Quadratic Function? Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. 6-5: Ex 1: Analyze Domain and Range & Try It! Curriculum Standards: Identify the function family when given an equation or graph. Identify the function family when given an equation or graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) Note: Emphasize the selection of a model function based on behavior of data and context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. Find the domain of a function defined symbolically, graphically or in a real-world context. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. 6-5: Additional Example 1 with Try Another One Curriculum Standards: Identify the function family when given an equation or graph. Identify the function family when given an equation or graph. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) Note: Emphasize the selection of a model function based on behavior of data and context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??(??) gives the number of person-hours it takes to assemble ?? engines in a factory, then the positive integers would be an appropriate domain for the function. Find the domain of a function defined symbolically, graphically or in a real-world context. Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. 6-5: Lesson Quiz (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. 6-5: Lesson Quiz Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. 6-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. end behavior; vertical and horizontal asymptotes; Analyze functions that include absolute value expressions. 6-5: Reteach to Build Understanding (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. 6-5: MathXL for School: Additional Practice Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Predict the behavior of polynomial functions. Analyze functions that include absolute value expressions. 6-5: Additional Practice (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. 6-5: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Analyze functions that include absolute value expressions. 6-5: Enrichment (PDF) Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. 6-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Identify the function family when given an equation or graph. 6-5: Virtual Nerd™: What is the Axis of Symmetry of a Quadratic Function? Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Translations of Functions Interactive Student Edition: Realize Reader: Lesson 6-6 Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Explore 6-6: Critique & Explain Understand and Apply 6-6: Ex 1: Vertical Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Ex 2: Analyze Horizontal Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Additional Example 2 Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Ex 3: Combine Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Concept Summary Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Do You Understand? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Do You Know How? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Practice and Problem Solving 6-6: MathXL for School: Practice and Problem Solving Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Assess & Differentiate 6-6: Ex 2: Analyze Horizontal Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Virtual Nerd™: What Does the Constant 'h' Do in the Function ??(??)=v(??-h)? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 6-6: MathXL for School: Enrichment Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 6-6: Ex 1: Vertical Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Additional Example 1 with Try Another One Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Ex 3: Combine Translations & Try It! Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Virtual Nerd™: What Does the Constant '??' Do in the Function ??(??)=v(??)+??? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Lesson Quiz (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Lesson Quiz Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 6-6: Reteach to Build Understanding (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: MathXL for School: Additional Practice Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-6: Additional Practice (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: MathXL for School: Enrichment Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Apply transformations to graph functions and write equations. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Apply transformations to graph functions and write equations. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. 6-6: Enrichment (PDF) Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. 6-6: Mathematical Literacy and Vocabulary (PDF) 6-6: Virtual Nerd™: What Does the Constant 'h' Do in the Function ??(??)=v(??-h)? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. 6-6: Virtual Nerd™: What Does the Constant '??' Do in the Function ??(??)=v(??)+??? Curriculum Standards: Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. Identify the effect on the graph of replacing ??(??) by ??(??) + ??, ?? ??(??), ??(????), and ??(?? + ??) for specific values of ?? (both positive and negative); find the value of ?? given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. Describe the effect of the transformations ????(??), ??(??)+??, ??(??+??), and combinations of such transformations on the graph of ??=??(??) for any real number ??. Find the value of ?? given the graphs and write the equation of a transformed parent function given its graph. Compressions and Stretches of Functions Interactive Student Edition: Realize Reader: Lesson 6-7 Curriculum Standards: Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Apply transformations to graph functions and write equations. Explore 6-7: Explore & Reason Understand and Apply 6-7: Ex 1: Analyze Reflections Across the ??-Axis & Try It! Curriculum Standards: Identify key features of the graph of the quadratic parent function. Change functions to compress or stretch their graphs. 6-7: Ex 2: Analyze Vertical Stretches of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Ex 3: Analyze Vertical Compressions of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Additional Example 3 Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Ex 4: Analyze Horizontal Stretches of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Additional Example 4 with Try Another One Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Ex 5: Analyze Horizontal Compressions of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Concept Summary Curriculum Standards: Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Apply transformations to graph functions and write equations. 6-7: Do You Understand? Curriculum Standards: Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Apply transformations to graph functions and write equations. 6-7: Do You Know How? Curriculum Standards: Change functions to compress or stretch their graphs. Identify key features of the graph of the quadratic parent function. Apply transformations to graph functions and write equations. Practice and Problem Solving 6-7: MathXL for School: Practice and Problem Solving Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. Assess & Differentiate 6-7: Ex 2: Analyze Vertical Stretches of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Additional Example 3 Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: MathXL for School: Enrichment Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Ex 4: Analyze Horizontal Stretches of Graphs & Try It! Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Additional Example 4 with Try Another One Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Lesson Quiz (PDF) Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Lesson Quiz Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Reteach to Build Understanding (PDF) Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: MathXL for School: Additional Practice Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: Additional Practice (PDF) Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. 6-7: MathXL for School: Enrichment Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Enrichment (PDF) Curriculum Standards: Change functions to compress or stretch their graphs. 6-7: Mathematical Literacy and Vocabulary (PDF) 6-7: Virtual Nerd™: What Does the Value of 'a' Do in the Function ??(??)=av(??)? Curriculum Standards: Change functions to compress or stretch their graphs. Apply transformations to graph functions and write equations. Operations With Functions Interactive Student Edition: Realize Reader: Lesson 6-8 Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Explore 6-8: Explore & Reason Understand and Apply 6-8: Ex 1: Add and Subtract Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: Ex 2: Multiply Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Additional Example 2 with Try Another One Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Ex 3: Apply Function Operations & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Additional Example 3 Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Concept Summary Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Do You Understand? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Do You Know How? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Practice and Problem Solving 6-8: MathXL for School: Practice and Problem Solving Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Assess & Differentiate 6-8: Ex 3: Apply Function Operations & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Ex 2: Multiply Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Ex 1: Add and Subtract Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: MathXL for School: Reteach to Build Understanding Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: Virtual Nerd™: How Do You Find the Product of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: Virtual Nerd™: How Do You Find the Sum of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Lesson Quiz (PDF) Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Lesson Quiz Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: MathXL for School: Reteach to Build Understanding Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: Reteach to Build Understanding (PDF) Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: MathXL for School: Additional Practice Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Additional Practice (PDF) Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: MathXL for School: Enrichment Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: Enrichment (PDF) Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Virtual Nerd™: How Do You Find the Sum of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. 6-8: Virtual Nerd™: How Do You Find the Product of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. Inverse Functions Interactive Student Edition: Realize Reader: Lesson 6-9 Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Explore 6-9: Explore & Reason Understand and Apply 6-9: Ex 1: Understand Inverse Functions & Try It! Curriculum Standards: Use inverse functions to solve problems. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Use inverse functions to solve problems. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (e.g., f(x) = 2 x³ or f(x) = (x+1)/(x-1) for x ? 1.) Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Extend this standard to simple rational, simple radical, and simple exponential functions; connect this standard to M.A2HS.34. Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f-_, and the range of the function f is the domain of the inverse function f-_. Solve an equation of the form ??(??) = ?? for a simple function ?? that has an inverse and write an expression for the inverse. Example: For example, ??(??) =2 ??³ or ??(??) = (??+1)/(??–1) for ?? ? 1. 6-9: Ex 2: Graph Inverse Functions & Try It! Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Additional Example 2 Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Ex 3: Find the Inverse of a Function Algebraically & Try It! Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Ex 4: Interpret Inverse Functions & Try It! Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Additional Example 4 with Try Another One Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Concept Summary Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. 6-9: Do You Understand? Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. 6-9: Do You Know How? Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Practice and Problem Solving 6-9: MathXL for School: Practice and Problem Solving Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Assess & Differentiate 6-9: Ex 2: Graph Inverse Functions & Try It! Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Additional Example 2 Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: MathXL for School: Enrichment Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Ex 1: Understand Inverse Functions & Try It! Curriculum Standards: Use inverse functions to solve problems. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Use inverse functions to solve problems. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; intercepts; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; end behavior; Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (e.g., f(x) = 2 x³ or f(x) = (x+1)/(x-1) for x ? 1.) Instructional Note: Use transformations of functions to find models as students consider increasingly more complex situations. Extend this standard to simple rational, simple radical, and simple exponential functions; connect this standard to M.A2HS.34. Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f-_, and the range of the function f is the domain of the inverse function f-_. Solve an equation of the form ??(??) = ?? for a simple function ?? that has an inverse and write an expression for the inverse. Example: For example, ??(??) =2 ??³ or ??(??) = (??+1)/(??–1) for ?? ? 1. 6-9: Lesson Quiz (PDF) Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. 6-9: Lesson Quiz Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. 6-9: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Reteach to Build Understanding (PDF) Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: MathXL for School: Additional Practice Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Additional Practice (PDF) Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: MathXL for School: Enrichment Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Enrichment (PDF) Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. 6-9: Virtual Nerd™: How Do You Find the Inverse of a Linear Function? Curriculum Standards: Use inverse functions to solve problems. Represent the inverse of a relation using tables, graphs, and equations. Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. Topic 6: MathXL for School: Topic Review Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Recognize the key features of exponential functions. Identify the function family when given an equation or graph. Topic 6: Performance Assessment Form A (PDF) Topic 6: Performance Assessment Form A Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Topic 6: Performance Assessment Form B Topic 6: Performance Assessment Form B (PDF) 6-2: Virtual Nerd™: What is a Piecewise Linear Function? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: Ex 4: Apply a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-8: Virtual Nerd™: How Do You Find the Product of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. end behavior; vertical and horizontal asymptotes; Analyze functions that include absolute value expressions. 6-3: MathXL for School: Enrichment Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-8: Ex 1: Add and Subtract Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: MathXL for School: Enrichment Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-5: Ex 2: Analyze Maximum and Minimum Values & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-3: Ex 2: Use a Step Function to Represent a Real-World Situation & Try It! Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-8: MathXL for School: Reteach to Build Understanding Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-3: Virtual Nerd™: What is a Step Function? Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Virtual Nerd™: How Do You Write an Equation for a Translation of an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-5: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Analyze functions that include absolute value expressions. 6-4: Ex 2: Vertical Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 1-3: Ex 5: Graph a Step Function & Try It! Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Topic 6: Assessment Form A (PDF) Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify the function family when given an equation or graph. Topic 6: Assessment Form A Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify the function family when given an equation or graph. Topic 6: Assessment Form B Topic 6: Assessment Form B (PDF) 6-2: Virtual Nerd™: What is a Piecewise Linear Function? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: Ex 4: Apply a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-8: Virtual Nerd™: How Do You Find the Product of Two Functions? Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. end behavior; vertical and horizontal asymptotes; Analyze functions that include absolute value expressions. 6-3: MathXL for School: Enrichment Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-8: Ex 1: Add and Subtract Functions & Try It! Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-8: MathXL for School: Enrichment Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-5: Ex 2: Analyze Maximum and Minimum Values & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-3: Ex 2: Use a Step Function to Represent a Real-World Situation & Try It! Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-8: MathXL for School: Reteach to Build Understanding Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. add, subtract, multiply, divide, and simplify rational algebraic expressions; composition of functions algebraically and graphically. Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.) Instructional Note: Develop models for more complex or sophisticated situations than in previous courses. Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Combine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations. 6-3: Virtual Nerd™: What is a Step Function? Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-4: Virtual Nerd™: How Do You Write an Equation for a Translation of an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-5: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Analyze functions that include absolute value expressions. 6-4: Ex 2: Vertical Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 6-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 1-3: Ex 5: Graph a Step Function & Try It! Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Topic 6: Assessment Form C Curriculum Standards: Add, subtract, and multiply functions. Perform operations on functions to answer real-world questions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Describe the key features of the square root function. Identify the key features of the cube root function. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify the function family when given an equation or graph. 6-2: Virtual Nerd™: What is a Piecewise Linear Function? Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-2: Ex 4: Apply a Piecewise-Defined Function & Try It! Curriculum Standards: Graph and apply piecewise-defined functions. Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Recognize the key features of exponential functions. end behavior; vertical and horizontal asymptotes; Analyze functions that include absolute value expressions. 6-5: Ex 2: Analyze Maximum and Minimum Values & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 6-5: MathXL for School: Enrichment Curriculum Standards: Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. domain and range; values of a function for elements in its domain; and connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Identify key features of quadratic functions. Write and graph quadratic functions in standard form. recognize the general shape of function families; and use knowledge of transformations to convert between equations and the corresponding graphs of functions. domain, range, and continuity; intervals in which a function is increasing or decreasing; extrema; zeros; intercepts; The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Analyze functions that include absolute value expressions. 6-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on quadratic functions; compare with linear and exponential functions studied in the Unit on Linear and Exponential Relationships. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Solve an equation of the form ??(??)=??(??) graphically by identifying the ??-coordinate(s) of the point(s) of intersection of the graphs of ??=??(??) and ??=??(??). (Limit to linear; quadratic; exponential.) Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.) Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form ??=????+??.) Graph and apply piecewise-defined functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Emphasize the selection of a model function based on behavior of data and context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Obtain information and draw conclusions from graphs of functions and other relations. Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f(x) = a(x – h)2 + k , or in factored form. Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. 5-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. 5-3: Ex 1: Solve Quadratic Equations & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: MathXL for School: Enrichment Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-3: Virtual Nerd™: How Do You Solve a Quadratic Equation With Complex Solutions by Using the Quadratic Formula? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. Mid-Year Assessment (PDF) Mid-Year Assessment Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. Find the zeros of quadratic functions. Use a variety of tools to solve systems of linear equations and inequalities. Solve systems of equations using matrices. Use matrices to represent and solve systems of equations. Model and solve problems using the zeros of a polynomial function. Describe and graph exponential functions. Perform, analyze, and use transformations of exponential functions. Identify the function family when given an equation or graph. Recognize the key features of exponential functions. Verify and use trigonometric identities. Write and graph quadratic functions in standard form. Factor a quadratic trinomial. Factor a quadratic trinomial when a ? 1. Predict the behavior of polynomial functions. Reason about operations with real numbers. Solve problems with complex numbers. Graph and apply piecewise-defined functions. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Analyze functions that include absolute value expressions. Identify key features of quadratic functions. Graph quadratic functions using the vertex form. Graph and analyze transformations of functions. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Relationships in Triangles 7-4: Virtual Nerd™: How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Perpendicular Line? Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 7-4: Ex 3: Check Perpendicularity & Try It! Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-3: Ex 2: Apply the SAS Congruence Criterion & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 6-1: Ex 5: Use the Angle Addition Postulate to Solve Problems & Try It! Curriculum Standards: Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; the perpendicular bisector of a line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Topic 7: Readiness Assessment (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Use properties of segments and angles to find their measures. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Topic 7: Readiness Assessment Curriculum Standards: Use slope to solve problems about parallel and perpendicular lines. Write equations of parallel lines and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use the midpoint and distance formulas to solve problems. Use properties of segments and angles to find their measures. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining the validity of a logical argument. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Topic 7: enVision STEM Project Topic 7: enVision STEM Project (PDF) Topic 7: enVision STEM Video Topic 7: enVision STEM Masters (PDF) Writing Proofs Interactive Student Edition: Realize Reader: Lesson 7-1 Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Explore 7-1: Critique & Explain Understand and Apply 7-1: Theorem 7-1 Vertical Angles Theorem 7-1: Ex 1: Write a Two-Column Proof & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Additional Example 1 Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Ex 2: Apply the Vertical Angles Theorem & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Additional Example 2 with Try Another One Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Theorem 7-2 Congruent Supplements Theorem 7-1: Theorem 7-3 Congruent Complements Theorem 7-1: Ex 3: Write a Paragraph Proof & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Theorem 7-4 7-1: Theorem 7-5 7-1: Theorem 7-6 Linear Pairs Theorem 7-1: Ex 4: Write a Proof Using a Theorem & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Concept Summary Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. 7-1: Do You Understand? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. 7-1: Do You Know How? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Practice and Problem Solving 7-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Assess & Differentiate 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-1: Ex 1: Write a Two-Column Proof & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Virtual Nerd™: What is a Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Lesson Quiz (PDF) Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-1: Lesson Quiz Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-1: Reteach to Build Understanding (PDF) Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-1: MathXL for School: Additional Practice Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. 7-1: Additional Practice (PDF) Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. 7-1: MathXL for School: Enrichment Curriculum Standards: Use deductive reasoning to prove theorems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Enrichment (PDF) Curriculum Standards: Use deductive reasoning to prove theorems. Identify different rigid motions used to transform two-dimensional shapes. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Mathematical Literacy and Vocabulary (PDF) 7-1: Virtual Nerd™: What is a Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Virtual Nerd™: What is the Vertical Angles Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Parallel Lines Interactive Student Edition: Realize Reader: Lesson 7-2 Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 7-2: Explore & Reason Understand and Apply 7-2: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Postulate 7-1 Same-Side Interior Angles Postulate 7-2: Ex 2: Explore Angle Relationships & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Additional Example 2 with Try Another One Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Theorem 7-7 Alternate Interior Angles Theorem 7-2: Theorem 7-8 Corresponding Angles Theorem 7-2: Theorem 7-9 Alternate Exterior Angles Theorem 7-2: Ex 3: Prove the Alternate Interior Angles Theorem & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 5: Find Angle Measures & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Additional Example 5 Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Concept Summary Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Do You Understand? Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Do You Know How? Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 7-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 7-2: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 5: Find Angle Measures & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Additional Example 5 Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Lesson Quiz (PDF) Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Lesson Quiz Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. 7-2: Reteach to Build Understanding (PDF) Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Additional Practice Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-2: Additional Practice (PDF) Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Enrichment (PDF) Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Virtual Nerd™: What is the Corresponding Angles Postulate? Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Virtual Nerd™: How Do You Find Missing Angles in a Transversal Diagram? Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Perpendicular and Angle Bisectors Interactive Student Edition: Realize Reader: Lesson 7-3 Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Explore 7-3: Model & Discuss Understand and Apply 7-3: Ex 1: Find Equidistant Points & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. 7-3: Theorem 7-10: Perpendicular Bisector Theorem 7-3: Theorem 7-11: Converse of the Perpendicular Bisector Theorem Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Ex 2: Prove the Perpendicular Bisector Theorem & Try-It! Curriculum Standards: Write conditionals and biconditionals and find their truth values. Use perpendicular and angle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Ex 3: Use a Perpendicular Bisector & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Ex 4: Apply the Perpendicular Bisector Theorem & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Additional Example 4 Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. 7-3: Ex 5: Find Equidistant Points from the Sides of an Angle & Try-Its! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Theorem 7-12: Angle Bisector Theorem 7-3: Theorem 7-13: Converse of the Angle Bisector Theorem 7-3: Ex 6: Apply the Angle Bisector Theorem & Try-Its! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Additional Example 6 with Try Another One Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Concept Summary Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. 7-3: Do You Understand? Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. 7-3: Do You Know How? Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Practice and Problem Solving 7-3: MathXL for School: Practice & Problem Solving Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. Use a straightedge and compass to construct basic figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. a line segment congruent to a given line segment; the bisector of a given angle, an angle congruent to a given angle; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Assess & Differentiate 6-4: Ex 3: Use a Conjecture to Make a Prediction & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 6-4: Virtual Nerd™: What is Inductive Reasoning? Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Identify and describe geometric sequences. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Instructional Note: Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. Recognize that geometric sequences are exponential using equations, tables, graphs and verbal descriptions. Given the formula f(x) = a(r)x, find the next term and define the meaning of a and r within the context of the problem. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Identify and describe geometric sequences. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 6-4: MathXL for School: Enrichment Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Lesson Quiz (PDF) Curriculum Standards: Use a straightedge and compass to construct basic figures. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; a line segment congruent to a given line segment; the perpendicular bisector of a line segment; the bisector of a given angle, an angle congruent to a given angle; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 7-3: Lesson Quiz Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Reteach to Build Understanding (PDF) Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: MathXL for School: Additional Practice Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Additional Practice (PDF) Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: MathXL for School: Enrichment Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. 7-3: Enrichment (PDF) Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use a straightedge and compass to construct basic figures. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; a line segment congruent to a given line segment; the perpendicular bisector of a line segment; the bisector of a given angle, an angle congruent to a given angle; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 7-3: Virtual Nerd™: How Do You Construct a Perpendicular Bisector? Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. 7-3: Virtual Nerd™: How Can You Tell if a Point is on the Perpendicular Bisector of a Line Segment? Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Write conditionals and biconditionals and find their truth values. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Bisectors in Triangles Interactive Student Edition: Realize Reader: Lesson 7-4 Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Explore 7-4: Model & Discuss Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 7-4: Theorem 7-14: Concurrency of Perpendicular Bisectors Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-4: Ex 1: Prove Theorem 7-14 & Try-It! Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Ex 2: Investigate Circumscribed Circles & Try-It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Ex 3: Use a Circumcenter & Try-It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Additional Example 3 Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Theorem 7-15: Concurrency of Angle Bisectors 7-4: Ex 4: Investigate Inscribed Circles & Try-It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-4: Ex 5: Identify and Use the Incenter of a Triangle Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Additional Example 5 with Try Another One Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Concept Summary Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Do You Understand? Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Do You Know How? Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 7-4: MathXL for School: Practice & Problem Solving Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Assess & Differentiate 7-4: Ex 2: Investigate Circumscribed Circles & Try-It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Ex 4: Investigate Inscribed Circles & Try-It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-4: Virtual Nerd™: What is an Incenter of a Triangle? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-4: Ex 5: Identify and Use the Incenter of a Triangle Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: MathXL for School: Enrichment Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Lesson Quiz (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: MathXL for School: Additional Practice Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 7-4: Additional Practice (PDF) Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: MathXL for School: Enrichment Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Enrichment (PDF) 7-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-4: Virtual Nerd™: What is an Incenter of a Triangle? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-4: Virtual Nerd™: What is the Circumcenter of a Triangle? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Mathematical Modeling in 3 Acts: Making it Fair Topic 7: Making it Fair - Act 1 Video with Questions Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 7: Making it Fair - Act 2 Content Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 7: Making it Fair - Act 2 Questions Topic 7: Making it Fair - Act 3 Video Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 7: Making it Fair - Act 3 Questions Medians and Altitudes Interactive Student Edition: Realize Reader: Lesson 7-5 Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; Explore 7-5: Critique & Explain Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; Understand and Apply 7-5: Ex 1: Identify Special Segments in Triangles & Try It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Theorem 7-16: Concurrency of Medians Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Ex 2: Find the Length of a Median & Try-It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Ex 3: Locate the Centroid & Try It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Additional Example 3 with Try Another One Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Theorem 7-17: Concurrency of Altitudes Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Ex 4: Locate the Orthocenter & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Ex 5: Find the Orthocenter of a Triangle & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Additional Example 5 Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Concept Summary Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; 7-5: Do You Understand? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; 7-5: Do You Know How? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; Practice and Problem Solving 7-5: MathXL for School: Practice & Problem Solving Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Solve problems using the measures of interior and exterior angles of triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. applying slope to verify and determine whether lines are parallel or perpendicular; The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Assess & Differentiate 7-5: Ex 2: Find the Length of a Median & Try-It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Lesson Quiz (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Lesson Quiz Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-5: Reteach to Build Understanding (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: MathXL for School: Additional Practice Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 7-5: Additional Practice (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: MathXL for School: Enrichment Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-5: Enrichment (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: Virtual Nerd™: What is a Median of a Triangle? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-5: Virtual Nerd™: How Do You Use the Centroid to Find Segment Lengths in a Triangle? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Inequalities in One Triangle Interactive Student Edition: Realize Reader: Lesson 7-6 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explore 7-6: Explore & Reason Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Understand and Apply 7-6: Ex 1: Investigate Side and Angle Relationships & Try It! Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Theorem 7-18 7-6: Ex 2: Use Theorem 7-18 & Try-It! Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Additional Example 2 Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Theorem 7-19: Converse of Theorem 7-18 7-6: Ex 3: Prove Theorem 7-19 & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Ex 4: Use Theorem 7-19 & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Theorem 7-20: Triangle Inequality Theorem Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Ex 5: Use the Triangle Inequality Theorem & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Additional Example 5 with Try Another One Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Concept Summary Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. The student, given information in the form of a figure or statement, will prove two triangles are congruent. 7-6: Do You Understand? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. The student, given information in the form of a figure or statement, will prove two triangles are congruent. 7-6: Do You Know How? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Practice and Problem Solving 7-6: MathXL for School: Practice & Problem Solving Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Assess & Differentiate 7-6: Ex 1: Investigate Side and Angle Relationships & Try It! Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Virtual Nerd™: How Do You Put the Sides of a Triangle in Order According to Size if You Know Two Angles of the Triangle? Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 7-6: MathXL for School: Enrichment Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Theorem 7-20: Triangle Inequality Theorem Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Virtual Nerd™: How Do You Determine Whether a Triangle Can Be Formed Given Three Side Lengths? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Lesson Quiz (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 7-6: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 7-6: Reteach to Build Understanding (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: MathXL for School: Additional Practice Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Additional Practice (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: MathXL for School: Enrichment Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Enrichment (PDF) Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Virtual Nerd™: How Do You Determine Whether a Triangle Can Be Formed Given Three Side Lengths? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Virtual Nerd™: How Do You Put the Sides of a Triangle in Order According to Size if You Know Two Angles of the Triangle? Curriculum Standards: Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. investigating and using formulas for determining distance, midpoint, and slope; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Inequalities in Two Triangles Interactive Student Edition: Realize Reader: Lesson 7-7 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Explore 7-7: Explore & Reason Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Understand and Apply 7-7: Ex 1: Investigate Side Lengths in Triangles & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Theorem 7-21: Hinge Theorem 7-7: Ex 2: Apply the Hinge Theorem & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Additional Example 2 Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Theorem 7-22: Converse of the Hinge Theorem 7-7: Ex 3: Prove the Converse of the Hinge Theorem & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-7: Ex 4: Apply the Converse of the Hinge Theorem & Try It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Additional Example 4 with Try Another One Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Concept Summary Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-7: Do You Understand? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-7: Do You Know How? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Practice and Problem Solving 7-7: MathXL for School: Practice & Problem Solving Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 7-6: Ex 4: Use Theorem 7-19 & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: Virtual Nerd™: How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: MathXL for School: Enrichment Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Theorem 7-20: Triangle Inequality Theorem Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Virtual Nerd™: How Do You Determine Whether a Triangle Can Be Formed Given Three Side Lengths? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 7-7: Lesson Quiz (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: Reteach to Build Understanding (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: MathXL for School: Additional Practice Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Additional Practice (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: MathXL for School: Enrichment Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: Enrichment (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Mathematical Literacy and Vocabulary (PDF) 7-7: Virtual Nerd™: What is the Hinge Theorem? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Virtual Nerd™: How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Topic 7: MathXL for School: Topic Review Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. determining the validity of a logical argument. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Topic 7: Performance Assessment: Form A (PDF) Curriculum Standards: Use a straightedge and compass to construct basic figures. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; a line segment congruent to a given line segment; the perpendicular bisector of a line segment; the bisector of a given angle, an angle congruent to a given angle; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Topic 7: Performance Assessment Form A Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Topic 7: Performance Assessment Form B Topic 7: Performance Assessment Form B (PDF) 7-3: Ex 1: Find Equidistant Points & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: Virtual Nerd™: How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Ex 4: Investigate Inscribed Circles & Try-It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. 7-1: Virtual Nerd™: What is the Vertical Angles Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-5: Virtual Nerd™: What is a Median of a Triangle? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: MathXL for School: Enrichment Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Solve problems using the measures of interior and exterior angles of triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 7-2: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Ex 2: Apply the Vertical Angles Theorem & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: MathXL for School: Enrichment Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 5: Find Angle Measures & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Ex 4: Use Theorem 7-10 & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Theorem 7-14: Concurrency of Perpendicular Bisectors Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Additional Example 4 Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. 7-2: Additional Example 5 Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-4: Virtual Nerd™: What is an Incenter of a Triangle? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. Topic 7: Assessment: Form A (PDF) Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Topic 7: Assessment Form A Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify and describe transformations of two-dimensional figures. Use triangle bisectors to solve problems. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use angle relationships to prove that lines are parallel. Use deductive reasoning to prove theorems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. determining the validity of a logical argument. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. Know and apply properties of a circle to solve problems and logically justify results. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Topic 7: Assessment Form B Topic 7: Assessment Form B (PDF) 7-3: Ex 1: Find Equidistant Points & Try It! Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. 7-7: Virtual Nerd™: How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-1: Virtual Nerd™: What is the Vertical Angles Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. 7-5: Virtual Nerd™: What is a Median of a Triangle? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-5: MathXL for School: Enrichment Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-2: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: Ex 2: Apply the Vertical Angles Theorem & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-7: MathXL for School: Enrichment Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-6: Ex 4: Use Theorem 7-19 & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-4: Theorem 7-14: Concurrency of Perpendicular Bisectors Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 7-2: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-3: Additional Example 4 Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. 7-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Topic 7: Assessment Form C Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify and describe transformations of two-dimensional figures. Use triangle bisectors to solve problems. Use angle relationships to prove that lines are parallel. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use properties of segments and angles to find their measures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Use deductive reasoning to prove theorems. Solve problems using the measures of interior and exterior angles of triangles. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. determining the validity of a logical argument. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Quadrilaterals and Other Polygons 9-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 9-2: Additional Example 3 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 9-2: Ex 3: Use the Converse of the Isosceles Triangle Theorem & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 9-6: Virtual Nerd™: How Do You Prove that Two Overlapping Triangles are Congruent? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. 7-5: Ex 2: Find the Length of a Median & Try-It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 7-1: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-6: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 7-1: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-6: Ex 1: Identify Corresponding Parts in Triangles & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 7-1: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Topic 8: Readiness Assessment (PDF) Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use deductive reasoning to prove theorems. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. applying slope to verify and determine whether lines are parallel or perpendicular; Topic 8: Readiness Assessment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. Use theorems to compare the sides and angles of a triangle. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. applying slope to verify and determine whether lines are parallel or perpendicular; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Topic 8: enVision STEM Project Topic 8: enVision STEM Project (PDF) Topic 8: enVision STEM Video Topic 8: enVision STEM Masters (PDF) The Polygon Angle-Sum Theorems Interactive Student Edition: Realize Reader: Lesson 8-1 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Explore 8-1: Explore & Reason Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Understand and Apply 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Additional Example 1 with Try Another One Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Theorem 8-1: Polygon Interior Angle-Sum 8-1: Corollary to Theorem 8-1 8-1: Ex 2: Apply the Polygon Interior Angle-Sum Theorem & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Additional Example 2 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Ex 3: Understand Exterior Angle Measures of a Polygon & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Theorem 8-2: Polygon Exterior Angle-Sum 8-1: Ex 4: Find an Exterior Angle Measure & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Ex 5: Find the Measures of Interior Angles & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Concept Summary Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Do You Understand? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Do You Know How? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Practice and Problem Solving 8-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Assess & Differentiate 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Lesson Quiz (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Lesson Quiz Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Reteach to Build Understanding (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: MathXL for School: Additional Practice Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Additional Practice (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Enrichment (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Virtual Nerd™: Why is the Sum of the Exterior Angles of a Polygon Always Equal to 360º? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Mathematical Modeling in 3 Acts: The Mystery Sides Topic 8: The Mystery Sides - Act 1 Video with Questions Topic 8: The Mystery Sides - Act 2 Content Topic 8: The Mystery Sides - Act 2 Questions Topic 8: The Mystery Sides - Act 3 Video Topic 8: The Mystery Sides - Act 3 Questions Kites and Trapezoids Interactive Student Edition: Realize Reader: Lesson 8-2 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Explore 8-2: Critique & Explain Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Understand and Apply 8-2: Ex 1: Investigate the Diagonals of a Kite & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Theorem 8-3 8-2: Ex 2: Use the Diagonals of a Kite & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Ex 3: Explore Parts of an Isosceles Trapezoid & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Theorem 8-4 8-2: Theorem 8-5 8-2: Ex 4: Solve Problems Involving Isosceles Trapezoids & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Additional Example 4 with Try Another One Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Theorem 8-6: Trapezoid Midsegment 8-2: Ex 5: Apply the Trapezoid Midsegment Theorem & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Additional Example 5 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Concept Summary Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Do You Understand? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Do You Know How? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Practice and Problem Solving 8-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Apply the properties of polygons to solve real-world and mathematical problems involving perimeter and area (e.g., triangles, special quadrilaterals, regular polygons up to 12 sides, composite figures). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Assess & Differentiate 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-2: Lesson Quiz (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Lesson Quiz Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Reteach to Build Understanding (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: MathXL for School: Additional Practice Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Additional Practice (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Enrichment (PDF) Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle). Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 8-2: Virtual Nerd™: How Do You Find a Value for a Variable in a Trapezoid? Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Properties of Parallelograms Interactive Student Edition: Realize Reader: Lesson 8-3 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 8-3: Critique & Explain Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 8-3: Ex 1: Explore Opposite Sides of Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Theorem 8-7 8-3: Ex 2: Use Opposite Sides of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Ex 3: Explore Angle Measures in Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Additional Example 3 with Try Another One Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Theorem 8-8 8-3: Theorem 8-9 8-3: Ex 4: Use Angles of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Theorem 8-10 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Additional Example 5 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Ex 6: Find Unknown Lengths in a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Concept Summary Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Do You Understand? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Do You Know How? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 8-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Assess & Differentiate 8-3: Ex 1: Explore Opposite Sides of Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Ex 3: Explore Angle Measures in Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Virtual Nerd™: How Do You Find Values for Variables in the Angles of a Quadrilateral To Make it a Parallelogram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Additional Example 5 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Lesson Quiz (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Lesson Quiz Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Reteach to Build Understanding (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: MathXL for School: Additional Practice Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Additional Practice (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: MathXL for School: Enrichment Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Enrichment (PDF) 8-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Virtual Nerd™: How Do You Find Values for Variables in the Angles of a Quadrilateral To Make it a Parallelogram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Proving a Quadrilateral Is a Parallelogram Interactive Student Edition: Realize Reader: Lesson 8-4 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 8-4: Explore & Reason Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 8-4: Ex 1: Investigate Sides to Confirm a Parallelogram & Try It! Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Theorem 8-11: Converse of Theorem 8-7 8-4: Ex 2: Explore Angle Measures to Confirm a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Theorem 8-12: Converse of Theorem 8-8 8-4: Theorem 8-13: Converse of Theorem 8-9 8-4: Ex 3: Find Values to Make Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Theorem 8-14: Converse of Theorem 8-10 8-4: Theorem 8-15 8-4: Ex 4: Investigate Diagonals to Confirm a Parallelogram & Try It! Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Additional Example 4 Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Ex 5: Identify a Parallelogram & Try It! Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Additional Example 5 with Try Another One Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Ex 6: Verify a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Concept Summary Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Do You Understand? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Do You Know How? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 8-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 8-4: Ex 1: Investigate Sides to Confirm a Parallelogram & Try It! Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Additional Example 4 Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: MathXL for School: Enrichment Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Ex 2: Explore Angle Measures to Confirm a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-3: Virtual Nerd™: How Do You Find Values for Variables in the Angles of a Quadrilateral To Make it a Parallelogram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-4: Lesson Quiz (PDF) Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Lesson Quiz Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Additional Practice Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Additional Practice (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-4: Enrichment (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Virtual Nerd™: How Do You Find the Values of Variables in a Parallelogram Diagram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Properties of Special Parallelograms Interactive Student Edition: Realize Reader: Lesson 8-5 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Explore 8-5: Explore & Reason Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Understand and Apply 8-5: Ex 1: Find the Diagonals of a Rhombus & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Theorem 8-16 8-5: Theorem 8-17 8-5: Ex 2: Find Lengths and Angle Measures in a Rhombus & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Additional Example 2 with Try Another One Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Theorem 8-18 8-5: Ex 3: Prove Diagonals of a Rectangle Are Congruent & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-5: Ex 4: Find Diagonal Lengths of a Rectangle & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Additional Example 4 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Ex 5: Diagonals and Angle Measures of a Square & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Concept Summary Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-5: Do You Understand? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-5: Do You Know How? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Practice and Problem Solving 8-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Assess & Differentiate 8-5: Ex 2: Find Lengths and Angle Measures in a Rhombus & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Find the Value for a Variable in the Angles of a Quadrilateral To Make it a Rhombus? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Use Variables to Name Coordinates for a Figure Placed on the Coordinate Plane? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Lesson Quiz (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Lesson Quiz Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Reteach to Build Understanding (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: MathXL for School: Additional Practice Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. 8-5: Additional Practice (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-5: MathXL for School: Enrichment Curriculum Standards: Use deductive reasoning to draw conclusions. Use angle relationships to prove that lines are parallel. Use perpendicular and angle bisectors to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Enrichment (PDF) 8-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Use Variables to Name Coordinates for a Figure Placed on the Coordinate Plane? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Find the Value for a Variable in the Angles of a Quadrilateral To Make it a Rhombus? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Conditions of Special Parallelograms Interactive Student Edition: Realize Reader: Lesson 8-6 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 8-6: Model & Discuss Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 8-6: Ex 1: Use Diagonals to Identify Rhombuses & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Theorem 8-19: Converse of Theorem 8-16 8-6: Theorem 8-20: Converse of Theorem 8-17 8-6: Ex 2: Prove the Parallelogram Diagonal Bisector Theorem & Try It! Curriculum Standards: Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Ex 3: Use Diagonals to Identify Rectangles & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Theorem 8-21: Converse of Theorem 8-18 8-6: Ex 4: Identify Special Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Additional Example 4 Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Ex 5: Use Properties of Special Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Additional Example 5 with Try Another One Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Ex 6: Apply Properties of Special Parallelograms & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Concept Summary Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Do You Understand? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Do You Know How? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 8-6: MathXL for School: Practice and Problem Solving Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Assess & Differentiate 8-6: Ex 3: Use Diagonals to Identify Rectangles & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: Virtual Nerd™: How Do You Find the Values of Variables in a Parallelogram Diagram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Virtual Nerd™: How Do You Use the Diagonals of a Rectangle to Find the Value of a Variable? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Lesson Quiz (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Lesson Quiz Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Reteach to Build Understanding (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: MathXL for School: Additional Practice Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-6: Additional Practice (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-6: Enrichment (PDF) 8-6: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Virtual Nerd™: How Do You Use the Diagonals of a Rectangle to Find the Value of a Variable? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 8: MathXL for School: Topic Review Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Topic 8: Performance Assessment Form A (PDF) Topic 8: Performance Assessment Form A Curriculum Standards: Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Topic 8: Performance Assessment Form B Topic 8: Performance Assessment Form B (PDF) 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-5: Ex 2: Find Lengths and Angle Measures in a Rhombus & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Ex 3: Use Diagonals to Identify Rectangles & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Find the Value for a Variable in the Angles of a Quadrilateral To Make it a Rhombus? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-4: Virtual Nerd™: How Do You Find the Values of Variables in a Parallelogram Diagram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Virtual Nerd™: How Do You Use the Diagonals of a Rectangle to Find the Value of a Variable? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 8: Assessment Form A (PDF) Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Topic 8: Assessment Form A Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Topic 8: Assessment Form B Topic 8: Assessment Form B (PDF) 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-5: Ex 2: Find Lengths and Angle Measures in a Rhombus & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Ex 3: Use Diagonals to Identify Rectangles & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-5: Virtual Nerd™: How Do You Find the Value for a Variable in the Angles of a Quadrilateral To Make it a Rhombus? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-4: MathXL for School: Enrichment Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Know and apply properties of congruent and similar figures to solve problems and logically justify results. 8-3: Ex 5: Explore the Diagonals of a Parallelogram & Try It! Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use the properties of rhombuses, rectangles, and squares to solve problems. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-4: Virtual Nerd™: How Do You Find the Values of Variables in a Parallelogram Diagram? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use properties of sides, angles, and diagonals to identify a parallelogram. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-6: Virtual Nerd™: How Do You Use the Diagonals of a Rectangle to Find the Value of a Variable? Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 8: Assessment Form C Curriculum Standards: Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Use the properties of rhombuses, rectangles, and squares to solve problems. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Use properties of sides, angles, and diagonals to identify a parallelogram. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 6-5: Ex 6: Identify the Conditionals in a Biconditional & Try It! Curriculum Standards: Write conditionals and biconditionals and find their truth values. Write conditionals and biconditionals and find their truth values. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Write conditionals and biconditionals and find their truth values. 7-3: Ex 3: Use the Triangle Angle-Sum Theorem & Try It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-4: Ex 5: Use the Geometric Mean to Solve Problems & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-3: Additional Example 3 with Try Another One Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 9-5: MathXL for School: Enrichment Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-6: Ex 4: Explore the Side Lengths of a 30°-60°-90° Triangle & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-7: Virtual Nerd™: How Do You Use the Hinge Theorem to Compare Side Lengths in Two Triangles? Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 6-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 9-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 6-4: MathXL for School: Enrichment Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 9-6: MathXL for School: Enrichment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-1: Additional Example 5 Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-5: Ex 5: Write and Evaluate a Biconditional & Try It! Curriculum Standards: Write conditionals and biconditionals and find their truth values. Write conditionals and biconditionals and find their truth values. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Write conditionals and biconditionals and find their truth values. 7-1: Ex 5: Find Angle Measures & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph and interpret piecewise-defined functions. domain, range, and continuity; connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 7-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply theorems about isosceles and equilateral triangles to solve problems. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. 9-6: Virtual Nerd™: How Do You Find Missing Sides in a 30º-60º-90º Triangle? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-6: Ex 3: Investigate Side Lengths in 45°-45°-90° Triangles & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-1: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 7-5: Virtual Nerd™: What is a Median of a Triangle? Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 6-3: MathXL for School: Enrichment Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 7-5: MathXL for School: Enrichment Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 9-4: Ex 1: Identify Similar Triangles Formed by an Altitude & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 6-7 MathXL for School: Reteach to Build Understanding Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-7 Virtual Nerd™: What is the Vertical Angles Theorem? Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-6: Ex 2: Use the Pythagorean Theorem and Its Converse & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 9-4: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 7-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use angle relationships to prove that lines are parallel. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use angle relationships to prove that lines are parallel. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use angle relationships to prove that lines are parallel. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-6: Additional Example 2 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 6-3: Ex 2: Use a Step Function to Represent a Real-World Situation & Try It! Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 7-2: Virtual Nerd™: How Do You Use Parallel and Perpendicular Theorems to Prove a Relationship Between Two Lines? Curriculum Standards: Use angle relationships to prove that lines are parallel. Use angle relationships to prove that lines are parallel. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use angle relationships to prove that lines are parallel. 7-7: MathXL for School: Enrichment Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 9-5: Additional Example 2 with Try Another One Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 7-1: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-7 Ex 2: Apply the Vertical Angles Theorem & Try It! Curriculum Standards: Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use deductive reasoning to prove theorems. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-3: Virtual Nerd™: What is a Step Function? Curriculum Standards: Graph and apply step functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Graph and apply step functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 7-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-6: Ex 4: Use Theorem 7-19 & Try-It! Curriculum Standards: Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. 7-4: Theorem 7-14: Concurrency of Perpendicular Bisectors Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Use triangle bisectors to solve problems. 6-4: Virtual Nerd™: How Do You Write an Equation for a Translation of an Absolute Value Function? Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 9-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Virtual Nerd™: What is a Geometric Mean? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 7-2: Ex 3: Determine Whether Lines Are Parallel & Try It! Curriculum Standards: Use angle relationships to prove that lines are parallel. Use angle relationships to prove that lines are parallel. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use angle relationships to prove that lines are parallel. 9-5: Ex 2: Use the Side-Splitter Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 7-1: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-4: Ex 2: Vertical Translations of the Absolute Value Function & Try It! Curriculum Standards: Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Recognize the graph of the functions f(x) = |x| and f(x) = x and predict the effects of transformations [f (x + c) and f(x) + c, where c is a positive or negative constant] algebraically and graphically using various methods and tools that may include graphing calculators. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Sketch the graphs of common non-linear functions such as f (x) = vx, f(x) = |x|, f(x) = 1/x, f(x) = x3 and translations of these functions, such as f(x) = vx-2+4. Know how to use graphing technology to graph these functions. 5-3: Ex 4: Interpret the Discriminant & Try It! Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Ex 4: Simplify a Quotient With Complex Numbers & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; 5-3: Virtual Nerd™: How Do You Find the Discriminant of a Quadratic Equation With 2 Complex Solutions? Curriculum Standards: Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Use the quadratic formula to solve quadratic equations. Solve quadratic equations using the Quadratic Formula. quadratic equations over the set of complex numbers; zeros; intercepts; The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. Solve quadratic equations by inspection (e.g., for ??² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ?? ± ???? for real numbers ?? and ??. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ??+???? for real numbers ?? and ??. 5-1: Ex 5: Factor a Sum of Squares & Try It! Curriculum Standards: Solve problems with complex numbers. Solve problems with complex numbers. The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. quadratic equations over the set of complex numbers; Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??² + 4 as (?? + 2??)(?? – 2??). Benchmark Test 3 (PDF) Benchmark Test 3 Curriculum Standards: Solve a system with linear and quadratic equations. Solve linear-quadratic systems. Write, graph, and apply the equation of a circle. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Solve problems with complex numbers. Identify and describe transformations of two-dimensional figures. Use triangle bisectors to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Graph and apply step functions. Use angle relationships to prove that lines are parallel. Analyze functions that include absolute value expressions. Graph and analyze transformations of the absolute value function. Write conditionals and biconditionals and find their truth values. Use deductive reasoning to prove theorems. Solve problems using the measures of interior and exterior angles of triangles. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use properties of exponents to solve equations with rational exponents. Write equivalent radical expressions. Relate roots and rational exponents and use them to simplify expressions and solve equations. Identify the function family when given an equation or graph. Use dilation and rigid motion to establish triangle similarity theorems. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Similarity and Right Triangles 9-3: Virtual Nerd™: What is CPCTC? Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 6-7: Virtual Nerd™: How Do You Identify a Similarity Transformation? Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 9-1: Virtual Nerd™: What Makes Two Figures Congruent? Curriculum Standards: Use a composition of rigid motions to show that two objects are congruent. Use a composition of rigid motions to show that two objects are congruent. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 7-2: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-3: Ex 2: Apply the SAS Congruence Criterion & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 7-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Use properties of segments and angles to find their measures. 6-5: Example 1 & Try It! Curriculum Standards: Use a composition of rigid motions to show that two objects are congruent. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use a composition of rigid motions to show that two objects are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 6-7: Example 3 & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 7-1: Ex 1: Identify Angle Pairs & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-4: MathXL for School: Enrichment Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 9-1: Ex 2: Verify Congruence & Try It! Curriculum Standards: Use a composition of rigid motions to show that two objects are congruent. Use a composition of rigid motions to show that two objects are congruent. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use a composition of rigid motions to show that two objects are congruent. Use a composition of rigid motions to show that two objects are congruent. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-3: Ex 4: Determine Congruent Triangles & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 6-7: Example 1 & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 7-1: Ex 4: Use Parallel Lines to Prove an Angle Relationship & Try It! Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 8-1: Ex 1: Identify Rigid Motions & Try It! Curriculum Standards: Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-1: MathXL for School: Enrichment Curriculum Standards: Use a composition of rigid motions to show that two objects are congruent. Solve problems using the measures of interior and exterior angles of triangles. Use a composition of rigid motions to show that two objects are congruent. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Solve problems using the measures of interior and exterior angles of triangles. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 7-1: MathXL for School: Enrichment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use angle relationships to prove that lines are parallel. Use the relationships between sides, segments, and angles of triangles to solve problems. 8-4: MathXL for School: Enrichment Curriculum Standards: Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Topic 9: Readiness Assessment (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Use angle relationships to prove that lines are parallel. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use a composition of rigid motions to show that two objects are congruent. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Topic 9: Readiness Assessment Curriculum Standards: Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Determine whether figures are similar. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Draw and describe the reflection of a figure across a line of reflection. Identify different rigid motions used to transform two-dimensional shapes. Dilate figures and identify characteristics of dilations. Use a composition of rigid motions to show that two objects are congruent. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Topic 9: enVision STEM Project Topic 9: enVision STEM Project (PDF) Topic 9: enVision STEM Video Topic 9: enVision STEM Masters (PDF) Dilations Interactive Student Edition: Realize Reader: Lesson 9-1 Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Explore 9-1: Explore & Reason Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Understand and Apply 9-1: Ex 1: Dilate a Figure & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. 9-1: Concept: Dilations 9-1: Ex 2: Analyze Dilations & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Ex 3: Find a Scale Factor & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Ex 4: Dilate a Figure With Center at the Origin & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Additional Example 4 with Try Another One Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Ex 5: Dilate a Figure With Center Not at the Origin & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Ex 6: Use a Scale Factor to Find Length and Area & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Additional Example 6 Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Do You Understand? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Do You Know How? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Concept Summary Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Practice and Problem Solving 9-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Apply the properties of polygons to solve real-world and mathematical problems involving perimeter and area (e.g., triangles, special quadrilaterals, regular polygons up to 12 sides, composite figures). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 9-1: Ex 2: Analyze Dilations & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Virtual Nerd™: How Do You Solve a Scale Model Problem Using a Scale Factor? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: Lesson Quiz (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Lesson Quiz Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: Reteach to Build Understanding (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: MathXL for School: Additional Practice Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Additional Practice (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: Enrichment (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Virtual Nerd™: How Do You Solve a Scale Model Problem Using a Scale Factor? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-1: Virtual Nerd™: How Do You Find a Scale Factor in Similar Figures? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Similarity Transformations Interactive Student Edition: Realize Reader: Lesson 9-2 Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. Explore 9-2: Critique & Explain Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Understand and Apply 9-2: Ex 1: Graph a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-2: Additional Example 1 with Try Another One Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-2: Ex 2: Describe a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Ex 3: Find Similarity Transformations & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Ex 4: Determine Similarity & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-2: Additional Example 4 Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-2: Ex 5: Identify Similar Circles & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: Concept Summary Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: Do You Understand? Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: Do You Know How? Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 9-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Assess & Differentiate 9-2: Ex 3: Find Similarity Transformations & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Virtual Nerd™: How Do You Identify a Similarity Transformation? Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: MathXL for School: Enrichment Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Ex 2: Describe a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Ex 5: Identify Similar Circles & Try It! Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: Ex 1: Graph a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-2: Virtual Nerd™: How Do You Graph a Translation Then a Dilation? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-2: Lesson Quiz (PDF) Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: Lesson Quiz Curriculum Standards: Determine whether figures are similar. Dilate figures and identify characteristics of dilations. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. 9-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Reteach to Build Understanding (PDF) Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: MathXL for School: Additional Practice Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Additional Practice (PDF) Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: MathXL for School: Enrichment Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Enrichment (PDF) Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-2: Virtual Nerd™: How Do You Graph a Translation Then a Dilation? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-2: Virtual Nerd™: How Do You Identify a Similarity Transformation? Curriculum Standards: Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Proving Triangles Similar Interactive Student Edition: Realize Reader: Lesson 9-3 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Explore 9-3: Explore & Reason Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Understand and Apply 9-3: Ex 1: Establish the Angle-Angle Similarity (AA ~) Theorem & Try It! Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Additional Example 1 with Try Another One Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Theorem 9-1: Angle-Angle Similarity (AA ~) Theorem 9-3: Ex 2: Establish the Side-Side-Side Similarity (SSS ~) Theorem & Try It! 9-3: Additional Example 2 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Ex 3: Verify Triangle Similarity & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Theorem 9-2: Side-Side-Side Similarity (SSS ~) Theorem 9-3: Theorem 9-3: Side-Angle-Side Similarity (SAS ~) Theorem 9-3: Ex 4: Find Lengths in Similar Triangles & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9-3: Ex 5: Solve Problems Involving Similar Triangles & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Concept Summary Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Do You Understand? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Do You Know How? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Practice and Problem Solving 9-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Assess & Differentiate 9-3: Ex 1: Establish the Angle-Angle Similarity (AA ~) Theorem & Try It! Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the AA Similarity Postulate? Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Additional Example 1 with Try Another One Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Ex 2: Establish the Side-Side-Side Similarity (SSS ~) Theorem & Try It! 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Lesson Quiz (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Reteach to Build Understanding (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: MathXL for School: Additional Practice Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: Additional Practice (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-3: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Enrichment (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the AA Similarity Postulate? Curriculum Standards: Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Similarity in Right Triangles Interactive Student Edition: Realize Reader: Lesson 9-4 Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and Explore 9-4: Explore & Reason Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and Understand and Apply 9-4: Ex 1: Identify Similar Triangles Formed by an Altitude & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Theorem 9-4 9-4: Ex 2: Find Missing Lengths Within Right Triangles & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Ex 3: Relate Altitude and Geometric Mean & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Corollary 1 to Theorem 9-4 9-4: Ex 4: Relate Side Lengths and Geometric Mean & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Addtional Example 4 with Try Another One Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 9-4: Corollary 2 to Theorem 9-4 9-4: Ex 5: Use the Geometric Mean to Solve Problems & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-4: Ex 6: Apply Geometric Mean to Find a Distance & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Additional Example 6 Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Concept Summary Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and 9-4: Do You Understand? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and 9-4: Do You Know How? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and Practice and Problem Solving 9-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Determine whether figures are similar. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Assess & Differentiate 9-4: Ex 1: Identify Similar Triangles Formed by an Altitude & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Lesson Quiz (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: MathXL for School: Additional Practice Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: Additional Practice (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Enrichment (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. properties of special right triangles; and Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-4: Virtual Nerd™: What is a Geometric Mean? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-4: Virtual Nerd™: How Do You Find a Geometric Mean? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Mathematical Modeling in 3 Acts: Make It Right Topic 9: Make It Right - Act 1 Video with Questions Topic 9: Make It Right - Act 2 Content Topic 9: Make It Right - Act 2 Questions Topic 9: Make It Right - Act 3 Video Topic 9: Make It Right - Act 3 Questions Proportions in Triangles Interactive Student Edition: Realize Reader: Lesson 9-5 Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 9-5: Explore & Reason Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 9-5: Ex 1: Explore Proportions from Parallel Lines & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Theorem 9-5: Side-Splitter Theorem 9-5: Theorem 9-6: Triangle Midsegment Theorem 9-5: Ex 2: Use the Side-Splitter Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Additional Example 2 with Try Another One Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Corollary to the Side-Splitter Theorem 9-5: Ex 3: Find a Length & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Ex 4: Investigate Proportionality with an Angle Bisector & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Theorem 9-7: Triangle-Angle-Bisector 9-5: Ex 5: Use the Triangle-Angle-Bisector Theorem & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Additional Example 5 Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Concept Summary Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Do You Understand? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Do You Know How? Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 9-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Assess & Differentiate 9-5: Ex 2: Use the Side-Splitter Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Additional Example 2 with Try Another One Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: MathXL for School: Enrichment Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Ex 1: Explore Proportions from Parallel Lines & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Virtual Nerd™: What is the Triangle Midsegment Theorem? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Ex 4: Investigate Proportionality with an Angle Bisector & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Lesson Quiz (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Lesson Quiz Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Reteach to Build Understanding (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: MathXL for School: Additional Practice Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. 9-5: Additional Practice (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: MathXL for School: Enrichment Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Enrichment (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-5: Virtual Nerd™: What is the Triangle Midsegment Theorem? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Virtual Nerd™: How Do You Use the Angle Bisector Theorem to Find Missing Side Lengths in a Diagram? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Right Triangles and the Pythagorean Theorem Interactive Student Edition: Realize Reader: Lesson 9-6 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 9-6: Explore & Reason Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Understand and Apply 9-6: Theorem 9-8: Pythagorean Theorem 9-6: Theorem 9-9: Converse of the Pythagorean Theorem 9-6: Ex 1: Use Similarity to Prove the Pythagorean Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; properties of special right triangles; and comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-6: Ex 2: Use the Pythagorean Theorem and Its Converse & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Additional Example 2 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Ex 3: Investigate Side Lengths in 45°-45°-90° Triangles & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Theorem 9-10: 45°-45°-90° Triangle Theorem 9-6: Ex 4: Explore the Side Lengths of a 30°-60°-90° Triangle & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Additional Example 4 with Try Another One Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-6: Theorem 9-11: 30°-60°-90° Triangle Theorem 9-6: Ex 5: Apply Special Right Triangle Relationships & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-6: Concept Summary Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-6: Do You Understand? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-6: Do You Know How? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 9-6: MathXL for School: Practice and Problem Solving Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 9-6: Ex 3: Investigate Side Lengths in 45°-45°-90° Triangles & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Virtual Nerd™: How Do You Find Missing Sides in a 30º-60º-90º Triangle? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Ex 4: Explore the Side Lengths of a 30°-60°-90° Triangle & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Ex 2: Use the Pythagorean Theorem and Its Converse & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Additional Example 2 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: MathXL for School: Enrichment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Lesson Quiz (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. 9-6: Lesson Quiz Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Reteach to Build Understanding (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-6: MathXL for School: Additional Practice Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. 9-6: Additional Practice (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. 9-6: MathXL for School: Enrichment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Enrichment (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-6: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 9-6: Virtual Nerd™: How Do You Find a Missing Hypotenuse in a 45º-45º-90º Triangle? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-6: Virtual Nerd™: How Do You Find Missing Sides in a 30º-60º-90º Triangle? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Trigonometric Ratios Interactive Student Edition: Realize Reader: Lesson 9-7 Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explore 9-7: Critique & Explain Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Understand and Apply 9-7: Concept: Trigonometric Ratios 9-7: Ex 1: Understand Trigonometric Ratios of Right Triangles Using Similarity & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Ex 2: Write Trigonometric Ratios & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Additional Example 2 with Try Another One 9-7: Ex 3: Trigonometric Ratios of Special Angles & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Ex 4: Use Trigonometric Ratios to Find Distances & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Additional Example 4 Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Ex 5: Use Trigonometric Inverses to Find Angle Measures & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Concept Summary Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Do You Understand? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Do You Know How? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Practice and Problem Solving 9-7: MathXL for School: Practice and Problem Solving Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Assess & Differentiate 9-7: Ex 1: Understand Trigonometric Ratios of Right Triangles Using Similarity & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Virtual Nerd™: What are Trigonometric Ratios? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Ex 3: Trigonometric Ratios of Special Angles & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Ex 4: Use Trigonometric Ratios to Find Distances & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Additional Example 4 Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: Ex 2: Write Trigonometric Ratios & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Additional Example 2 with Try Another One 9-7: Ex 5: Use Trigonometric Inverses to Find Angle Measures & Try It! Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Virtual Nerd™: What are the Values of the Trigonometric Ratios in a 30º-60º-90º Triangle? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Lesson Quiz (PDF) Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: Lesson Quiz Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. 9-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: Reteach to Build Understanding (PDF) Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: MathXL for School: Additional Practice Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: Additional Practice (PDF) Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: MathXL for School: Enrichment Curriculum Standards: Identify, evaluate, and graph linear functions. 9-7: Enrichment (PDF) 9-7: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9-7: Virtual Nerd™: What are Trigonometric Ratios? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9-7: Virtual Nerd™: What are the Values of the Trigonometric Ratios in a 30º-60º-90º Triangle? Curriculum Standards: Use trigonometric ratios to find lengths and angle measures of right triangles. trigonometric ratios. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Extension 9-7a: The Pythagorean Identity Interactive Student Edition: Realize Reader: Lesson 9-7a 9-7a: Ex 1: Prove the Pythagorean Identity & Try-It 9-7a: Ex 2: Use the Pythagorean Identity & Try-It 9-7a: Do You Understand 9-7a: Do You Understand9-7a: Do You Understand 9-7a: Do You Know How 9-7a: Do You Know How9-7a: Do You Know How 9-7a: Concept Summary 9-7a: MathXL for School: Practice & Problem Solving Topic 9: MathXL for School: Topic Review Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine whether figures are similar. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Dilate figures and identify characteristics of dilations. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. solving problems, including practical problems, about similar geometric figures. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers. Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. Analyze the effects on summary statistics of changes in data sets. Topic 9: Performance Assessment Form A (PDF) Topic 9: Performance Assessment Form A Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Topic 9: Performance Assessment Form B Topic 9: Performance Assessment Form B (PDF) 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-5: MathXL for School: Enrichment Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Ex 1: Explore Proportions from Parallel Lines & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-6: MathXL for School: Enrichment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-2: Virtual Nerd™: How Do You Graph a Translation Then a Dilation? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Ex 1: Identify Similar Triangles Formed by an Altitude & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-5: Ex 4: Investigate Proportionality with an Angle Bisector & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Virtual Nerd™: What is the Triangle Midsegment Theorem? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-6: Ex 2: Use the Pythagorean Theorem and Its Converse & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-2: Ex 1: Graph a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-4: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-6: Additional Example 2 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-5: Additional Example 2 with Try Another One Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-5: Ex 2: Use the Side-Splitter Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Topic 9: Assessment Form A (PDF) Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine whether figures are similar. Dilate figures and identify characteristics of dilations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. Topic 9: Assessment Form A Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine whether figures are similar. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use trigonometric ratios to find lengths and angle measures of right triangles. Use the Law of Sines to solve problems. Dilate figures and identify characteristics of dilations. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. trigonometric ratios. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Explain and use the relationship between the sine and cosine of complementary angles. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. (HONORS ONLY) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Topic 9: Assessment Form B Topic 9: Assessment Form B (PDF) 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-5: MathXL for School: Enrichment Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Ex 1: Explore Proportions from Parallel Lines & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-2: Virtual Nerd™: How Do You Graph a Translation Then a Dilation? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-3: Virtual Nerd™: How Do You Determine if Two Triangles are Similar Using the SAS Similarity Postulate? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-4: Ex 1: Identify Similar Triangles Formed by an Altitude & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-5: Ex 4: Investigate Proportionality with an Angle Bisector & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-5: Virtual Nerd™: What is the Triangle Midsegment Theorem? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-2: Ex 1: Graph a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-4: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-5: Additional Example 2 with Try Another One Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. Use similarity and the geometric mean to solve problems involving right triangles. 9-5: Ex 2: Use the Side-Splitter Theorem & Try It! Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Topic 9: Assessment Form C Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine whether figures are similar. Dilate figures and identify characteristics of dilations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use dilation and rigid motion to establish triangle similarity theorems. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Know and apply properties of congruent and similar figures to solve problems and logically justify results. solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove that all circles are similar. Prove that all circles are similar. Prove that all circles are similar. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of a circle to solve problems and logically justify results. Probability 7-5: Example 1 & Try It! Curriculum Standards: Define probability distributions to represent experiments and solve problems. Define probability distributions to represent experiments and solve problems. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 7-4: Example 1 & Try It! Curriculum Standards: Define probability distributions to represent experiments and solve problems. Define probability distributions to represent experiments and solve problems. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. 7-2: Example 3 & Try It! Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? Topic 10: Readiness Assessment (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Topic 10: Readiness Assessment Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Use the properties of prisms and cylinders to calculate their volumes. Use inductive reasoning to make conjectures about mathematical relationships. Use perpendicular and angle bisectors to solve problems. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. The student will use surface area and volume of three-dimensional objects to solve practical problems. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. determining the validity of a logical argument. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; The student will verify and use properties of quadrilaterals to solve problems, including practical problems. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Topic 10: enVision STEM Project Topic 10: enVision STEM Project (PDF) Topic 10: enVision STEM Video Topic 10: enVision STEM Masters (PDF) Two-Way Frequency Tables Interactive Student Edition: Realize Reader: Lesson 10-1 Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Explore 10-1: Explore & Reason Understand and Apply 10-1: Ex 1: Interpret a Two-Way Frequency Table & Try It! Curriculum Standards: Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Additional Example 1 Curriculum Standards: Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Ex 2: Interpret a Two-Way Relative Frequency Table & Try It! Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Ex 3: Calculate Conditional Relative Frequency & Try It! Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Ex 4: Interpret Conditional Relative Frequency & Try It! Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Ex 5: Interpret Data Frequencies & Try It! Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Additional Example 5 with Try Another One Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Concept Summary Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Do You Understand? Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Do You Know How? Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Practice and Problem Solving 10-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Assess & Differentiate 1-1: Ex 1: Solve Linear Equations & Try It! 1-1: Ex 2: Solve Consecutive Integer Problems & Try It! 1-1: MathXL for School: Reteach to Build Understanding 1-1: Virtual Nerd™: How Do You Solve a Two-Step Equation? 1-2: Ex 2: Understand Equations With Infinitely Many and No Solutions & Try It! 1-2: Ex 3: Solve Mixture Problems & Try It! 1-2: MathXL for School: Enrichment 1-2: Virtual Nerd™: How Do You Solve a Word Problem Using an Equation With Variables on Both Sides? 4-4: Ex 4: Determine a Reasonable Solution & Try It! Curriculum Standards: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 10-1: Lesson Quiz (PDF) Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Lesson Quiz Curriculum Standards: Create and solve linear equations with one variable. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. 10-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Reteach to Build Understanding (PDF) Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: MathXL for School: Additional Practice Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Additional Practice (PDF) Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: MathXL for School: Enrichment Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Enrichment (PDF) Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. 10-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-1: Virtual Nerd™: How Do You Find Relative Frequency? Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Probability Events Interactive Student Edition: Realize Reader: Lesson 10-2 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Explore 10-2: Explore & Reason Understand and Apply 10-2: Ex 1: Find Probabilities of Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Additional Example 1 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Concept: Probabilities of Mutually Exclusive Events 10-2: Ex 2: Find the Probabilities of Non-Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Concept: Probabilities of Non-Mutually Exclusive Events 10-2: Ex 3: Identify Independent Events & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Concept: Probability of Independent Events 10-2: Ex 4: Find Probabilities of Independent Events & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Additional Example 4 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Concept Summary Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Do You Understand? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Do You Know How? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Practice and Problem Solving 10-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Assess & Differentiate 10-2: Ex 3: Identify Independent Events & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Ex 4: Find Probabilities of Independent Events & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Additional Example 4 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Ex 1: Find Probabilities of Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of Independent Events? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Lesson Quiz Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Additional Practice (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Enrichment (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. 10-2: Virtual Nerd™: How Do You Find the Probability of Independent Events? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of the Complement of an Event? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Conditional Probability Interactive Student Edition: Realize Reader: Lesson 10-3 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Explore 10-3: Explore & Reason Understand and Apply 10-3: Ex 1: Understand Conditional Probability & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Additional Example 1 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Concept: Conditional Probability and Independent Events 10-3: Ex 2: Use the Test for Independence & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Additional Example 2 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 3: Apply the Conditional Probability Formula & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 4: Use Conditional Probability to Make a Decision & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Concept Summary Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Do You Understand? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Do You Know How? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Practice and Problem Solving 10-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Assess & Differentiate 10-3: Ex 2: Use the Test for Independence & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find a Conditional Probability Using a Contingency Table? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 1: Understand Conditional Probability & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find Conditional Probability? Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Lesson Quiz Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Additional Practice (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Enrichment (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find Conditional Probability? Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find a Conditional Probability Using a Contingency Table? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Mathematical Modeling in 3 Acts: Place Your Guess Topic 10: Place Your Guess - Act 1 Video with Questions Topic 10: Place Your Guess - Act 2 Content Topic 10: Place Your Guess - Act 2 Questions Topic 10: Place Your Guess - Act 3 Video Topic 10: Place Your Guess - Act 3 Questions Permutations and Combinations Interactive Student Edition: Realize Reader: Lesson 10-4 Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Explore 10-4: Explore & Reason Understand and Apply 10-4: Ex 1: Use the Fundamental Counting Principle & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Ex 2: Find the Number of Permutations & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Additional Example 2B Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Concept: Permutations 10-4: Ex 3: Find the Number of Combinations & Try It! Curriculum Standards: Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. The student will compute and distinguish between permutations and combinations. (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Concept: Combinations 10-4: Ex 4: Use Permutations and Combinations to Find Probabilities & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Concept Summary Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Do You Understand? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Do You Know How? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Practice and Problem Solving 10-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Assess & Differentiate 10-4: Ex 3: Find the Number of Combinations & Try It! Curriculum Standards: Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. The student will compute and distinguish between permutations and combinations. (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Ex 2: Find the Number of Permutations & Try It! Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Virtual Nerd™: How Do You Solve a Word Problem Using the Permutation Formula? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Lesson Quiz Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Additional Practice (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Enrichment (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Virtual Nerd™: What is the Permutation Formula? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-4: Virtual Nerd™: How Do You Solve a Word Problem Using the Permutation Formula? Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Probability Distributions Interactive Student Edition: Realize Reader: Lesson 10-5 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Explore 10-5: Model & Discuss Understand and Apply 10-5: Ex 1: Develop a Theoretical Probability Distribution & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Ex 2: Develop an Experimental Probability Distribution & Try It! Curriculum Standards: Define probability distributions to represent experiments and solve problems. Define probability distributions to represent experiments and solve problems. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Concept: Binomial Experiments 10-5: Ex 3: Binomial Experiments & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Additional Example 3 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Concept: Binomial Probability Formula 10-5: Ex 4: Probabilities in a Binomial Experiment & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Additional Example 4 with Try Another One Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Concept Summary Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Do You Understand? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Do You Know How? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Practice and Problem Solving 10-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Assess & Differentiate 10-5: Ex 1: Develop a Theoretical Probability Distribution & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Virtual Nerd™: How Do You Compare Actual Results to Predicted Results? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Ex 3: Binomial Experiments & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Additional Example 3 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Lesson Quiz Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Additional Practice (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Enrichment (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Mathematical Literacy and Vocabulary (PDF) 10-5: Virtual Nerd™: How Do You Compare Actual Results to Predicted Results? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-5: Virtual Nerd™: How Do You Use a Simulation to Solve a Problem? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Expected Value Interactive Student Edition: Realize Reader: Lesson 10-6 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Explore 10-6: Explore & Reason Understand and Apply 10-6: Ex 1: Evaluate and Apply Expected Value & Try It! Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Ex 2: Find Expected Payoffs & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Additional Example 2 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Ex 3: Use Expected Values to Evaluate Strategies & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Ex 4: Use Binomial Probability to Find Expected Value & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Concept Summary Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Do You Understand? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Do You Know How? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Practice and Problem Solving 10-6: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Assess & Differentiate 10-6: Ex 3: Use Expected Values to Evaluate Strategies & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Ex 4: Use Binomial Probability to Find Expected Value & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Ex 1: Evaluate and Apply Expected Value & Try It! Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Virtual Nerd™: How Do You Find an Expected Value? Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Ex 2: Find Expected Payoffs & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Additional Example 2 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Lesson Quiz Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Additional Practice (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Enrichment (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-6: Virtual Nerd™: How Do You Find an Expected Value? Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Virtual Nerd™: What is Expected Value? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Probability and Decision Making Interactive Student Edition: Realize Reader: Lesson 10-7 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Explore 10-7: Critique & Explain Understand and Apply 10-7: Ex 1: Use Probability to Make Fair Decisions & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Ex 2: Determine Whether a Decision Is Fair or Unfair & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Additional Example 2 Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Ex 3: Make a Decision Based on Expected Value & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Ex 4: Use a Binomial Distribution to Make Decisions & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Concept Summary Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Do You Understand? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Do You Know How? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Practice and Problem Solving 10-7: MathXL for School: Practice and Problem Solving Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Assess & Differentiate 10-7: Ex 1: Use Probability to Make Fair Decisions & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Virtual Nerd™: How can you use probability to make decisions? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Ex 3: Make a Decision Based on Expected Value & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Ex 4: Use a Binomial Distribution to Make Decisions & Try It! Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Lesson Quiz (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Lesson Quiz Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Reteach to Build Understanding (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: MathXL for School: Additional Practice Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Additional Practice (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: MathXL for School: Enrichment Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Enrichment (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. 10-7: Virtual Nerd™: How can you use probability to make decisions? Curriculum Standards: Apply probability concepts to real-world situations to make informed decisions. Topic 10: MathXL for School: Topic Review Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Topic 10: Performance Assessment Form A (PDF) Topic 10: Performance Assessment Form A Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Topic 10: Performance Assessment Form B Topic 10: Performance Assessment Form B (PDF) 10-6: Virtual Nerd™: How Do You Find an Expected Value? Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-2: Ex 2: Find the Probabilities of Non-Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find Conditional Probability? Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of the Complement of an Event? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 3: Apply the Conditional Probability Formula & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 1: Understand Conditional Probability & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of Independent Events? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 4: Use Conditional Probability to Make a Decision & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-6: Ex 1: Evaluate and Apply Expected Value & Try It! Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-4: Ex 3: Find the Number of Combinations & Try It! Curriculum Standards: Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. The student will compute and distinguish between permutations and combinations. (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-5: Ex 2: Develop an Experimental Probability Distribution & Try It! Curriculum Standards: Define probability distributions to represent experiments and solve problems. Define probability distributions to represent experiments and solve problems. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-4: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Ex 1: Find Probabilities of Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Topic 10: Assessment Form A (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Topic 10: Assessment Form A Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Topic 10: Assessment Form B Topic 10: Assessment Form B (PDF) Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-6: Virtual Nerd™: How Do You Find an Expected Value? Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-2: Ex 2: Find the Probabilities of Non-Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Virtual Nerd™: How Do You Find Conditional Probability? Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of the Complement of an Event? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 3: Apply the Conditional Probability Formula & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 1: Understand Conditional Probability & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Recognize the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Instructional Note: Build on work with two-way tables from Algebra I to develop understanding of conditional probability and independence. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Virtual Nerd™: How Do You Find the Probability of Independent Events? Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-3: Ex 4: Use Conditional Probability to Make a Decision & Try It! Curriculum Standards: Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Find the probability of an event given that another event has occurred. Find the probability of an event given that another event has occurred. Find the conditional probability of ?? given ?? as the fraction of ??’s outcomes that also belong to ??, and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-6: Ex 1: Evaluate and Apply Expected Value & Try It! Curriculum Standards: Calculate, interpret, and apply expected value. Calculate, interpret, and apply expected value. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-4: Ex 3: Find the Number of Combinations & Try It! Curriculum Standards: Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use permutations and combinations to find the number of outcomes in a probability experiment. Use permutations and combinations to find the number of outcomes in a probability experiment. The student will compute and distinguish between permutations and combinations. (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-5: Ex 2: Develop an Experimental Probability Distribution & Try It! Curriculum Standards: Define probability distributions to represent experiments and solve problems. Define probability distributions to represent experiments and solve problems. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 10-4: Additional Example 3 with Try Another One Curriculum Standards: Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. 10-2: Ex 1: Find Probabilities of Mutually Exclusive Events & Try It! Curriculum Standards: Use relationships among events to find probabilities. Use relationships among events to find probabilities. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Use relationships among events to find probabilities. Use relationships among events to find probabilities. (HONORS ONLY) Apply the Addition Rule, ??(?? or ??) = ??(??) + ??(??) – ??(?? and ??), and interpret the answer in terms of the model. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Topic 10: Assessment Form C Curriculum Standards: Organize data in two-way frequency tables and use them to make inferences and generalizations. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. Understand and use simple probability formulas involving intersections, unions and complements of events. Apply probability concepts to real-world situations to make informed decisions. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 9-3: Virtual Nerd™: What is CPCTC? Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 6-1: Ex 6: Use Congruent Angles and Congruent Segments & Try It! Curriculum Standards: Use properties of segments and angles to find their measures. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; the perpendicular bisector of a line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use properties of segments and angles to find their measures. 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-5: Ex 1: Explore Proportions from Parallel Lines & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 5-5: Ex 4: Expand a Power of a Binomial & Try It! Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 6-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of segments and angles to find their measures. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; the perpendicular bisector of a line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Use properties of segments and angles to find their measures. 9-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 5-5: Ex 3: Use Polynomial Identities to Factor and Simplify & Try It! Curriculum Standards: Prove and use polynomial identities. 9-3: Ex 2: Apply the SAS Congruence Criterion & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 6-1: MathXL for School: Enrichment Curriculum Standards: Use properties of segments and angles to find their measures. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; the perpendicular bisector of a line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use properties of segments and angles to find their measures. 9-2: Virtual Nerd™: How Do You Graph a Translation Then a Dilation? Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-5: Ex 4: Investigate Proportionality with an Angle Bisector & Try It! Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 9-4: MathXL for School: Enrichment Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 9-5: Virtual Nerd™: What is the Triangle Midsegment Theorem? Curriculum Standards: Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. 5-5: MathXL for School: Enrichment Curriculum Standards: Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. 9-2: Ex 1: Graph a Composition of a Rigid Motion and a Dilation & Try It! Curriculum Standards: Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 9-3: Ex 4: Determine Congruent Triangles & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Determine congruent triangles by comparing two angles and one side. 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 6-1: Ex 5: Use the Angle Addition Postulate to Solve Problems & Try It! Curriculum Standards: Use properties of segments and angles to find their measures. Use properties of segments and angles to find their measures. a line segment congruent to a given line segment; the perpendicular bisector of a line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use properties of segments and angles to find their measures. Benchmark Test 4 (PDF) Benchmark Test 4 Curriculum Standards: Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Use the relationships between sides, segments, and angles of triangles to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use angle relationships to prove that lines are parallel. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. Prove and use polynomial identities. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Analyze functions that include absolute value expressions. Identify the function family when given an equation or graph. Use dilation and rigid motion to establish triangle similarity theorems. Use SAS and SSS to determine whether triangles are congruent. Use properties of segments and angles to find their measures. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays. Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. a line segment congruent to a given line segment; an angle congruent to a given angle; Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Coordinate Geometry 6-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. 7-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: While functions will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra I. (e.g., Finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve linear programming problems in two variables using graphical methods. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. 7-4: Ex 1: Slopes of Parallel Lines & Try It! Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 7-4: Virtual Nerd™: How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Perpendicular Line? Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-3: Ex 4: Find the Distance & Try It! Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. 6-3: Ex 1: Find a Midpoint & Try It! Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the midpoint and distance formulas to solve problems. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the midpoint and distance formulas to solve problems. 6-3: MathXL for School: Enrichment Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the midpoint and distance formulas to solve problems. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the midpoint and distance formulas to solve problems. 7-4: Virtual Nerd™: How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Parallel Line? Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 7-4: Ex 3: Check Perpendicularity & Try It! Curriculum Standards: Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. represent verbal quantitative situations algebraically; and evaluate algebraic expressions for given replacement values of the variables. determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. 6-3: Additional Example 4 Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. 7-4: Ex 4: Write Equations of Parallel and Perpendicular Lines & Try It! Curriculum Standards: Use slope to solve problems about parallel and perpendicular lines. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) Use slope to solve problems about parallel and perpendicular lines. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use slope to solve problems about parallel and perpendicular lines. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: While functions will often be linear, exponential, or quadratic the types of problems should draw from more complex situations than those addressed in Algebra I. (e.g., Finding the equation of a line through a given point perpendicular to another line allows one to find the distance from a point to a line). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve linear programming problems in two variables using graphical methods. Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. 6-3: Virtual Nerd™: How Do You Find the Midpoint Between Two Coordinates? Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the midpoint and distance formulas to solve problems. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the midpoint and distance formulas to solve problems. Topic 11: Readiness Assessment (PDF) Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use slope to solve problems about parallel and perpendicular lines. Write equations of parallel lines and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the equations and graphs of circles to solve problems. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Topic 11: Readiness Assessment Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use slope to solve problems about parallel and perpendicular lines. Write equations of parallel lines and perpendicular lines. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the equations and graphs of circles to solve problems. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Topic 11: enVision STEM Project Topic 11: enVision STEM Project (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Topic 11: enVision STEM Video Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Topic 11: enVision STEM Masters (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Polygons in the Coordinate Plane Interactive Student Edition: Realize Reader: Lesson 11-1 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Explore 11-1: Explore & Reason Understand and Apply 11-1: Ex 1: Connect Algebra and Geometry Through Coordinates & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Additional Example 1 with Try Another Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: Ex 2: Classify a Triangle on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Ex 3: Classify a Parallelogram on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-1: Ex 4: Classify Quadrilaterals as Trapezoids and Kites on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Ex 5: Find Perimeter and Area & Try It! Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the coordinate plane to analyze geometric figures. 11-1: Additional Example 5 Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-1: Concept Summary Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: Do You Understand? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: Do You Know How? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Practice and Problem Solving 11-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Assess & Differentiate 11-1: Ex 1: Connect Algebra and Geometry Through Coordinates & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Additional Example 5 Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove geometric theorems using algebra and the coordinate plane. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Ex 5: Find Perimeter and Area & Try It! Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the coordinate plane to analyze geometric figures. 11-1: Additional Example 1 with Try Another Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: Ex 3: Classify a Parallelogram on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-1: Virtual Nerd™: How Do You Find the Area of a Parallelogram on the Coordinate Plane? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-1: Lesson Quiz (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove geometric theorems using algebra and the coordinate plane. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: Reteach to Build Understanding (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: MathXL for School: Additional Practice Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: Additional Practice (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. 11-1: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Enrichment (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-1: Virtual Nerd™: How Do You Find the Area of a Parallelogram on the Coordinate Plane? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Mathematical Modeling in 3 Acts: You Be the Judge Topic 11: You Be the Judge - Act 1 Video with Questions Topic 11: You Be the Judge - Act 2 Content Topic 11: You Be the Judge - Act 2 Questions Topic 11: You Be the Judge - Act 3 Video Topic 11: You Be the Judge - Act 3 Questions Proofs Using Coordinate Geometry Interactive Student Edition: Realize Reader: Lesson 11-2 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Explore 11-2: Critique & Explain Understand and Apply 11-2: Ex 1: Plan a Coordinate Proof & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Additional Example 1 Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Ex 2: Write a Coordinate Proof & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: Ex 3: Plan and Write a Coordinate Proof & Try It! Curriculum Standards: Identify and describe transformations of two-dimensional figures. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: Ex 4: Use Coordinate Proofs to Solve Problems & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Additional Example 4 with Try Another One Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Concept Summary Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: Do You Understand? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: Do You Know How? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Practice and Problem Solving 11-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Assess & Differentiate 11-2: Ex 2: Write a Coordinate Proof & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: Virtual Nerd™: How Do You Write a Coordinate Proof? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 11-2: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: Lesson Quiz (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Lesson Quiz Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. 11-2: Reteach to Build Understanding (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: MathXL for School: Additional Practice Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: Additional Practice (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. 11-2: MathXL for School: Enrichment Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: Enrichment (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 11-2: Virtual Nerd™: How Do You Write a Coordinate Proof? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-2: Virtual Nerd™: How Do You Position a Figure on the Coordinate Plane for a Coordinate Proof? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Circles in the Coordinate Plane Interactive Student Edition: Realize Reader: Lesson 11-3 Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the midpoint and distance formulas to solve problems. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Explore 11-3: Model & Discuss Understand and Apply 11-3: Ex 1: Derive the Equation of a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Theorem 11-1: Equation of a Circle 11-3: Ex 2: Write the Equation of a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. 11-3: Ex 3: Determine Whether a Point Lies on a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-3: Additional Example 3 with Try Another One Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-3: Ex 4: Graph a Circle from Its Equation & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. 11-3: Ex 5: Use the Graph and Equation of a Circle to Solve Problems & Try It! Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-3: Additional Example 5 Curriculum Standards: Use the midpoint and distance formulas to solve problems. Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-3: Concept Summary Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the midpoint and distance formulas to solve problems. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-3: Do You Understand? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the midpoint and distance formulas to solve problems. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-3: Do You Know How? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the midpoint and distance formulas to solve problems. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Practice and Problem Solving 11-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the midpoint and distance formulas to solve problems. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Assess & Differentiate 11-3: Ex 3: Determine Whether a Point Lies on a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-3: Additional Example 3 with Try Another One Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Ex 1: Derive the Equation of a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Virtual Nerd™: How Do You Derive the Equation For a Circle? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Lesson Quiz (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: Lesson Quiz Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Reteach to Build Understanding (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: MathXL for School: Additional Practice Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: Additional Practice (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: Enrichment (PDF) 11-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 11-3: Virtual Nerd™: How Do You Derive the Equation For a Circle? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-3: Virtual Nerd™: How Do You Graph a Circle Without Making a Table? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Parabolas in the Coordinate Plane Interactive Student Edition: Realize Reader: Lesson 11-4 Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Explore 11-4: Explore & Reason Understand and Apply 11-4: Ex 1: Explore the Graph of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: Ex 2: Derive the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-4: Concept: Equation of a Parabola 11-4: Ex 3: Write the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Additional Example 3 with Try Another One Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Ex 4: Apply the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Additional Example 4 Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Concept Summary Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Do You Understand? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Do You Know How? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Practice and Problem Solving 11-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Assess & Differentiate 11-4: Ex 2: Derive the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: What is a Parabola? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. Understand and apply the geometric properties of a parabola. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Understand and apply the geometric properties of a parabola. 11-4: Ex 1: Explore the Graph of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: How Does the Equation of a Vertical Parabola Relate to Its Graph? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: MathXL for School: Enrichment Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: Lesson Quiz (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Lesson Quiz Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. Understand and apply the geometric properties of a parabola. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Understand and apply the geometric properties of a parabola. 11-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: MathXL for School: Additional Practice Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Additional Practice (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: MathXL for School: Enrichment Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: Enrichment (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: How Does the Equation of a Vertical Parabola Relate to Its Graph? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: What is a Parabola? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Topic 11: MathXL for School: Topic Review Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Use the equations and graphs of parabolas to solve problems. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Understand and apply the geometric properties of a parabola. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Topic 11: Performance Assessment Form A (PDF) Topic 11: Performance Assessment Form A Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Topic 11: Performance Assessment Form B Topic 11: Performance Assessment Form B (PDF) 11-3: Additional Example 3 with Try Another One Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-4: Virtual Nerd™: What is a Parabola? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: How Does the Equation of a Vertical Parabola Relate to Its Graph? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-3: Ex 3: Determine Whether a Point Lies on a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-2: Virtual Nerd™: How Do You Write a Coordinate Proof? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-1: Virtual Nerd™: How Do You Find the Area of a Parallelogram on the Coordinate Plane? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-4: Ex 1: Explore the Graph of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-2: Ex 2: Write a Coordinate Proof & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-3: Virtual Nerd™: How Do You Derive the Equation For a Circle? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-4: Ex 2: Derive the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove geometric theorems using algebra and the coordinate plane. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: Additional Example 1 with Try Another Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. Understand and apply the geometric properties of a parabola. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Understand and apply the geometric properties of a parabola. 11-3: Ex 1: Derive the Equation of a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-1: Ex 3: Classify a Parallelogram on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-1: Ex 5: Find Perimeter and Area & Try It! Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the coordinate plane to analyze geometric figures. Topic 11: Assessment Form A (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Topic 11: Assessment Form A Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Topic 11: Assessment Form B Topic 11: Assessment Form B (PDF) Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 11-3: Additional Example 3 with Try Another One Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-4: Virtual Nerd™: What is a Parabola? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-4: Virtual Nerd™: How Does the Equation of a Vertical Parabola Relate to Its Graph? Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-3: Ex 3: Determine Whether a Point Lies on a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. The student will solve problems involving equations of circles. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. 11-2: Virtual Nerd™: How Do You Write a Coordinate Proof? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-1: Virtual Nerd™: How Do You Find the Area of a Parallelogram on the Coordinate Plane? Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-4: Ex 1: Explore the Graph of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. 11-2: Ex 2: Write a Coordinate Proof & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-3: Virtual Nerd™: How Do You Derive the Equation For a Circle? Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-4: Ex 2: Derive the Equation of a Parabola & Try It! Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Derive the equation of a parabola given a focus and directrix. 11-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove geometric theorems using algebra and the coordinate plane. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-1: Additional Example 1 with Try Another Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. 11-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. Understand and apply the geometric properties of a parabola. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use the equations and graphs of parabolas to solve problems. Derive the equation of a parabola given a focus and directrix. Understand and apply the geometric properties of a parabola. 11-3: Ex 1: Derive the Equation of a Circle & Try It! Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically. 11-1: Ex 3: Classify a Parallelogram on the Coordinate Plane & Try It! Curriculum Standards: Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. 11-1: Ex 5: Find Perimeter and Area & Try It! Curriculum Standards: Use the coordinate plane to analyze geometric figures. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Use the coordinate plane to analyze geometric figures. Topic 11: Assessment Form C Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use the relationships between sides, segments, and angles of triangles to solve problems. Use the coordinate plane to analyze geometric figures. Prove geometric theorems using algebra and the coordinate plane. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. investigating and using formulas for determining distance, midpoint, and slope; applying slope to verify and determine whether lines are parallel or perpendicular; Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to prove simple geometric theorems algebraically. Assess the validity of a logical argument and give counterexamples to disprove a statement. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Circles 9-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Apply theorems about isosceles and equilateral triangles to solve problems. Use the relationships between sides, segments, and angles of triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-2: Additional Example 3 Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-2: Ex 3: Use the Converse of the Isosceles Triangle Theorem & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; The student, given information in the form of a figure or statement, will prove two triangles are congruent. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Apply theorems about isosceles and equilateral triangles to solve problems. Identify congruent right triangles. Use theorems to compare the sides and angles of a triangle. 9-6: Ex 4: Explore the Side Lengths of a 30°-60°-90° Triangle & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: MathXL for School: Enrichment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: MathXL for School: Reteach to Build Understanding Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-6: Virtual Nerd™: How Do You Find Missing Sides in a 30º-60º-90º Triangle? Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-6: Ex 3: Investigate Side Lengths in 45°-45°-90° Triangles & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-3: Ex 1: Explore the Side-Angle-Side (SAS) Congruence Criterion & Try It! Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-3: MathXL for School: Enrichment Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-6: Ex 2: Use the Pythagorean Theorem and Its Converse & Try It! Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. 9-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. The student, given information in the form of a figure or statement, will prove two triangles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Use SAS and SSS to determine whether triangles are congruent. Use the relationships between sides, segments, and angles of triangles to solve problems. 9-6: Additional Example 2 Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Topic 12: Readiness Assessment (PDF) Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use SAS and SSS to determine whether triangles are congruent. Use theorems to compare the sides and angles of a triangle. Find arc length and sector area of a circle and use them to solve problems. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Topic 12: Readiness Assessment Curriculum Standards: Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use SAS and SSS to determine whether triangles are congruent. Use theorems to compare the sides and angles of a triangle. Find arc length and sector area of a circle and use them to solve problems. properties of special right triangles; and Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Topic 12: enVision STEM Project Topic 12: enVision STEM Project (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Find arc length and sector area of a circle and use them to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Topic 12: enVision STEM Video Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Find arc length and sector area of a circle and use them to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Topic 12: enVision STEM Masters (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Find arc length and sector area of a circle and use them to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Arcs and Sectors Interactive Student Edition: Realize Reader: Lesson 12-1 Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Explore 12-1: Explore & Reason Understand and Apply 12-1: Ex 1: Relate Central Angles and Arc Measures & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Concept: Arc Measure 12-1: Ex 2: Relate Arc Length to Circumference & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Concept: Arc Length 12-1: Ex 3: Apply Arc Length & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Additional Example 3 with Try Another One Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Ex 4: Relate the Area of a Circle to the Area of a Sector & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 5: Find the Area of a Segment of a Circle & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Additional Example 5 Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Ex 6: Solve Problems Involving Circles & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Concept Summary Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Do You Understand? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Do You Know How? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 12-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Assess & Differentiate 12-1: Ex 1: Relate Central Angles and Arc Measures & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: MathXL for School: Enrichment Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Ex 4: Relate the Area of a Circle to the Area of a Sector & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Virtual Nerd™: What is the Formula for the Area of a Sector of a Circle? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 3: Apply Arc Length & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Virtual Nerd™: What is the Formula for Arc Length? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 2: Relate Arc Length to Circumference & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Lesson Quiz (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Lesson Quiz Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Reteach to Build Understanding (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: MathXL for School: Additional Practice Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Additional Practice (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: MathXL for School: Enrichment Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Enrichment (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Virtual Nerd™: What is the Formula for Arc Length? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Virtual Nerd™: What is the Formula for the Area of a Sector of a Circle? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. Lines Tangent to a Circle Interactive Student Edition: Realize Reader: Lesson 12-2 Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Explore 12-2: Critique & Explain Understand and Apply 12-2: Ex 1: Understand Tangents to a Circle & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Theorem 12-1 and the Converse 12-2: Ex 2: Use Tangents to Solve Problems & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Additional Example 2 with Try Another One Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Ex 3: Find Lengths of Segments Tangent to a Circle & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Theorem 12-2: Segments Tangent to a Circle 12-2: Ex 4: Find Measures Involving Tangent Lines & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Additional Example 4 Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Ex 5: Construct Tangent Lines & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Concept Summary Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Do You Understand? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Do You Know How? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 12-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Assess & Differentiate 12-2: Ex 1: Understand Tangents to a Circle & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Virtual Nerd™: What is a Tangent Line to a Circle? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Enrichment Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Lesson Quiz (PDF) Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Lesson Quiz Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Reteach to Build Understanding (PDF) Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Additional Practice Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Additional Practice (PDF) Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Enrichment Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Enrichment (PDF) Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Virtual Nerd™: How Do You Determine Whether a Line is Tangent to a Circle? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: Virtual Nerd™: What is a Tangent Line to a Circle? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Mathematical Modeling in 3 Acts: Earth Watch Topic 12: Earth Watch - Act 1 Video with Questions Topic 12: Earth Watch - Act 2 Content Topic 12: Earth Watch - Act 2 Questions Topic 12: Earth Watch - Act 3 Video Topic 12: Earth Watch - Act 3 Questions Chords Interactive Student Edition: Realize Reader: Lesson 12-3 Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Explore 12-3: Explore & Reason Understand and Apply 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Theorem 12-3 And The Converse 12-3: Theorem 12-4 And The Converse 12-3: Ex 2: Relate Arcs and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Theorem 12-5 And The Converse 12-3: Ex 3: Relate Chords Equidistant from the Center & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Additional Example 3 with Try Another One Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 12-3: Ex 4: Construct a Regular Hexagon Inscribed in a Circle & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 12-3: Theorem 12-6 And The Converse 12-3: Theorem 12-7 12-3: Ex 5: Solve Problems Involving Chords of Circles & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Additional Example 5 Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Concept Summary Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 12-3: Do You Understand? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 12-3: Do You Know How? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Practice and Problem Solving 12-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Assess & Differentiate 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Lesson Quiz (PDF) Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Lesson Quiz Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Reteach to Build Understanding (PDF) Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: MathXL for School: Additional Practice Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Additional Practice (PDF) Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Enrichment (PDF) Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Virtual Nerd™: How Do You Find the Length of a Chord in a Circle if You're Given another Chord Equidistant from the Center? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Inscribed Angles Interactive Student Edition: Realize Reader: Lesson 12-4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Explore 12-4: Explore & Reason Understand and Apply 12-4: Ex 1: Relate Inscribed Angles to Intercepted Arcs & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Theorem 12-8: Inscribed Angles 12-4: Ex 2: Use the Inscribed Angles Theorem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Corollaries to the Inscribed Angles Theorem 12-4: Ex 3: Explore Angles Formed by a Tangent and a Chord & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Additional Example 3 with Try Another One Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Theorem 12-9 12-4: Ex 4: Use Arc Measure to Solve a Problem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Additional Example 4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Concept Summary Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Do You Understand? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Do You Know How? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 12-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Assess & Differentiate 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-4: Ex 2: Use the Inscribed Angles Theorem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Ex 4: Use Arc Measure to Solve a Problem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Additional Example 4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 1: Relate Inscribed Angles to Intercepted Arcs & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Virtual Nerd™: How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. 12-4: Lesson Quiz (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Lesson Quiz Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Reteach to Build Understanding (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and 12-4: MathXL for School: Additional Practice Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and 12-4: Additional Practice (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and 12-4: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. 12-4: Enrichment (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Virtual Nerd™: How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Virtual Nerd™: How Do You Find Missing Measures of Angles in Quadrilaterals Inscribed in Circles? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Secant Lines and Segments Interactive Student Edition: Realize Reader: Lesson 12-5 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Explore 12-5: Explore & Reason Understand and Apply 12-5: Ex 1: Relate Secants and Angle Measures & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Theorem 12-10 12-5: Theorem 12-11 12-5: Ex 2: Prove Theorem 10-11, Case 1 & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Ex 3: Use Secants and Tangents to Solve Problems & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Additional Example 3A with Try Another One Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Ex 4: Develop Chord Length Relationships & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Theorem 12-12 12-5: Ex 5: Use Segment Relationships to Find Lengths & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Additional Example 5 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Concept Summary Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Do You Understand? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Do You Know How? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Practice and Problem Solving 12-5: MathXL for School: Practice and Problem Solving Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Assess & Differentiate 12-4: Ex 4: Use Arc Measure to Solve a Problem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Additional Example 4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Ex 3: Use Secants and Tangents to Solve Problems & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 1: Relate Inscribed Angles to Intercepted Arcs & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: Virtual Nerd™: How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. 12-5: Lesson Quiz (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Lesson Quiz Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Reteach to Build Understanding (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: MathXL for School: Additional Practice Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Additional Practice (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Enrichment (PDF) Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Mathematical Literacy and Vocabulary (PDF) 12-5: Virtual Nerd™: How Do You Use Intersecting Chords to Find Arc Measures in a Circle? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-5: Virtual Nerd™: How Do You Find the Measure of an Angle Created by Intersecting Chords in a Circle? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Topic 12: MathXL for School: Topic Review Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Find arc length and sector area of a circle and use them to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Topic 12: Performance Assessment Form A (PDF) Topic 12: Performance Assessment Form A Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. Use properties of tangent lines to solve problems. Use the Law of Sines to solve problems. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. trigonometric ratios. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. (HONORS ONLY) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Topic 12: Performance Assessment Form B Topic 12: Performance Assessment Form B (PDF) 12-1: Ex 2: Relate Arc Length to Circumference & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-2: MathXL for School: Enrichment Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 4: Use Arc Measure to Solve a Problem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Virtual Nerd™: What is the Formula for Arc Length? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-5: Ex 3: Use Secants and Tangents to Solve Problems & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 1: Relate Inscribed Angles to Intercepted Arcs & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-4: Virtual Nerd™: How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-1: Ex 3: Apply Arc Length & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-4: Additional Example 4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Virtual Nerd™: What is the Formula for the Area of a Sector of a Circle? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 4: Relate the Area of a Circle to the Area of a Sector & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-4: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. 12-2: Ex 1: Understand Tangents to a Circle & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-2: Virtual Nerd™: What is a Tangent Line to a Circle? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Topic 12: Assessment Form A (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Topic 12: Assessment Form A Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Topic 12: Assessment Form B Topic 12: Assessment Form B (PDF) Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. 12-1: Ex 2: Relate Arc Length to Circumference & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-2: MathXL for School: Enrichment Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 4: Use Arc Measure to Solve a Problem & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Virtual Nerd™: What is the Formula for Arc Length? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-3: Virtual Nerd™: How Do You Determine Whether Two Chords are Equidistant From the Center of a Circle? Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-5: Ex 3: Use Secants and Tangents to Solve Problems & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-4: Ex 1: Relate Inscribed Angles to Intercepted Arcs & Try It! Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-3: MathXL for School: Enrichment Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-4: Virtual Nerd™: How Do You Find The Measure of an Inscribed Angle When You Know the Measure of the Intercepted Arc? Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. 12-1: Ex 3: Apply Arc Length & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-4: Additional Example 4 Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. 12-1: Virtual Nerd™: What is the Formula for the Area of a Sector of a Circle? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 4: Relate the Area of a Circle to the Area of a Sector & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-4: MathXL for School: Enrichment Curriculum Standards: Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. 12-2: Ex 1: Understand Tangents to a Circle & Try It! Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-3: Ex 1: Relate Central Angles and Chords & Try It! Curriculum Standards: Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. 12-2: Virtual Nerd™: What is a Tangent Line to a Circle? Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. 12-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use properties of tangent lines to solve problems. angle measures formed by intersecting chords, secants, and/or tangents; Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Know and apply properties of a circle to solve problems and logically justify results. Topic 12: Assessment Form C Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. Use relationships between circles, angles, and arcs. Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use properties of tangent lines to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. Construct a tangent line from a point outside a given circle to the circle. (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. Two- and Three-Dimensional Models 8-1: Virtual Nerd™: How Do You Find the Sum of the Interior Angles of a Polygon Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 9-1: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. comparing ratios between lengths, perimeters, areas, and volumes of similar figures; solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. 8-1: Ex 1: Explore Polygon Interior Angle Sums & Try It! Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-3: Example 1 & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use the volumes of right and oblique pyramids and cones to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 12-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 8-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 8-1: MathXL for School: Enrichment Curriculum Standards: Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. 12-1: Virtual Nerd™: What is the Formula for the Area of a Sector of a Circle? Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 12-1: Ex 4: Relate the Area of a Circle to the Area of a Sector & Try It! Curriculum Standards: Find arc length and sector area of a circle and use them to solve problems. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Find arc length and sector area of a circle and use them to solve problems. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Find arc length and sector area of a circle and use them to solve problems. 8-3: Virtual Nerd™: How Do You Find the Volume of a Cone? Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use the volumes of right and oblique pyramids and cones to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 8-2: Example 1 & Try It! 8-2: Example 1 & Try It!This animated component is the first part of the Visual Learning Bridge from the student edition. Some use interactivity to illustrate math ideas. It is designed for whole-class instruction. Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. 8-2: Virtual Nerd™: How Do You Find the Volume of a Cylinder? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Topic 13: Readiness Assessment (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the volumes of right and oblique pyramids and cones to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Find arc length and sector area of a circle and use them to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Topic 13: Readiness Assessment Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the volumes of right and oblique pyramids and cones to solve problems. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Find arc length and sector area of a circle and use them to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. arc length; and area of a sector. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. Know and apply properties of a circle to solve problems and logically justify results. Topic 13: enVision STEM Project Topic 13: enVision STEM Project (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. Topic 13: enVision STEM Video Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. Topic 13: enVision STEM Masters (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. Three-Dimensional Figures and Cross Sections Interactive Student Edition: Realize Reader: Lesson 13-1 Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Explore 13-1: Explore & Reason Understand and Apply 13-1: Ex 1: Develop Euler’s Formula & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Concept: Euler’s Formula 13-1: Ex 2: Apply Euler’s Formula & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Additional Example 2 with Try Another One Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 3: Describe a Cross Section & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Additional Example 3 Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 4: Draw a Cross Section & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 5: Rotate a Polygon to Form a Three-Dimensional Figure & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Concept Summary Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Do You Understand? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Do You Know How? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Practice and Problem Solving 13-1: MathXL for School: Practice and Problem Solving Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Assess & Differentiate 13-1: Ex 3: Describe a Cross Section & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Virtual Nerd™: How Do You List the Vertices, Edges, and Faces of a Polyhedron? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: MathXL for School: Enrichment Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 5: Rotate a Polygon to Form a Three-Dimensional Figure & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Virtual Nerd™: What is a Solid? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Lesson Quiz (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Lesson Quiz Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Reteach to Build Understanding (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: MathXL for School: Additional Practice Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Additional Practice (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: MathXL for School: Enrichment Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Enrichment (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Virtual Nerd™: How Do You List the Vertices, Edges, and Faces of a Polyhedron? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Virtual Nerd™: What is a Solid? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Volumes of Prisms and Cylinders Interactive Student Edition: Realize Reader: Lesson 13-2 Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explore 13-2: Model & Discuss Understand and Apply 13-2: Ex 1: Develop Cavalieri’s Principle & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-2: Concept: Cavalieri’s Principle 13-2: Concept: Volumes of Prisms and Cylinders 13-2: Ex 2: Find the Volumes of Prisms and Cylinders & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Ex 3: Apply the Volumes of Prisms to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Additional Example 3 Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Ex 4: Apply Volume of Cylinders to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Additional Example 4 with Try Another One Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Ex 5: Determine Whether Volume or Surface Area Best Describes Size & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. The student will use surface area and volume of three-dimensional objects to solve practical problems. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-2: Concept Summary Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Do You Understand? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Do You Know How? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Use geometric shapes, their measures, and their properties to describe real-world objects. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Practice and Problem Solving 13-2: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Assess & Differentiate 13-2: Ex 3: Apply the Volumes of Prisms to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Additional Example 3 Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Lesson Quiz (PDF) Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Lesson Quiz Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Reteach to Build Understanding (PDF) Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Additional Practice Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Additional Practice (PDF) Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Enrichment (PDF) Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Virtual Nerd™: What is the Formula for the Volume of a Prism? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: Virtual Nerd™: How Do You Find the Volume of a Cylinder? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Mathematical Modeling in 3 Acts: Box 'Em Up Topic 13: Box 'Em Up - Act 1 Video with Questions Topic 13: Box 'Em Up - Act 2 Content Topic 13: Box 'Em Up - Act 2 Questions Topic 13: Box 'Em Up - Act 3 Video Topic 13: Box 'Em Up - Act 3 Questions Pyramids and Cones Interactive Student Edition: Realize Reader: Lesson 13-3 Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explore 13-3: Explore & Reason Understand and Apply 13-3: Ex 1: Apply Cavalieri’s Principle to Pyramids and Cones & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-3: Concept: Volumes of Pyramids and Cones 13-3: Ex 2: Find the Volumes of Pyramids and Cones & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 3: Apply the Volumes of Pyramids to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Example 3 with Try Another One Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 4: Apply the Volumes of Cones to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Example 4 Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 5: Measure a Composite Figure & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Concept Summary Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Do You Understand? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Do You Know How? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Practice and Problem Solving 13-3: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Assess & Differentiate 13-3: Ex 3: Apply the Volumes of Pyramids to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Example 3 with Try Another One Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Enrichment Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 2: Find the Volumes of Pyramids and Cones & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 5: Measure a Composite Figure & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Virtual Nerd™: How Do You Find the Volume of a Composite Figure? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Lesson Quiz (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Lesson Quiz Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Reteach to Build Understanding (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Additional Practice Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Practice (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Enrichment Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Enrichment (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Virtual Nerd™: How Do You Find the Volume of a Composite Figure? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Virtual Nerd™: What is the Formula for the Volume of a Cone? Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Spheres Interactive Student Edition: Realize Reader: Lesson 13-4 Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Explore 13-4: Critique & Explain Understand and Apply 13-4: Ex 1: Explore the Volume of a Sphere & Try It! Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Concept: Volume of a Sphere 13-4: Ex 2: Use the Volumes of Spheres to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Additional Example 2 Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 3: Find the Volumes of Hemispheres & Try It! Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Additional Example 3 with Try Another One Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 4: Find the Volumes of Composite Figures & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Concept Summary Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Do You Understand? Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Do You Know How? Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Practice and Problem Solving 13-4: MathXL for School: Practice and Problem Solving Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Assess & Differentiate 13-4: Ex 3: Find the Volumes of Hemispheres & Try It! Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Additional Example 3 with Try Another One Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Enrichment Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 2: Use the Volumes of Spheres to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Virtual Nerd™: How Do You Find the Volume of a Composite Figure? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Lesson Quiz (PDF) Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Lesson Quiz Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Reteach to Build Understanding (PDF) Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Additional Practice Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Additional Practice (PDF) Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Enrichment Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Enrichment (PDF) Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Mathematical Literacy and Vocabulary (PDF) Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Virtual Nerd™: How Do You Find the Volume of a Sphere? Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Virtual Nerd™: What is a Sphere? Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Topic 13: MathXL for School: Topic Review Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Identify space figures and their relationships with polygons to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Topic 13: Performance Assessment Form A (PDF) Topic 13: Performance Assessment Form A Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Topic 13: Performance Assessment Form B Topic 13: Performance Assessment Form B (PDF) 13-3: Ex 4: Apply the Volumes of Cones to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Enrichment Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: MathXL for School: Enrichment Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 3: Describe a Cross Section & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 3: Apply the Volumes of Pyramids to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 2: Use the Volumes of Spheres to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 3: Find the Volumes of Hemispheres & Try It! Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Enrichment Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Ex 5: Rotate a Polygon to Form a Three-Dimensional Figure & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Virtual Nerd™: What is a Solid? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: Virtual Nerd™: How Do You Find the Volume of a Composite Figure? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: Ex 1: Apply Cavalieri’s Principle to Pyramids and Cones & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-2: Ex 1: Develop Cavalieri’s Principle & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-3: Virtual Nerd™: What is the Formula for the Volume of a Cone? Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Virtual Nerd™: How Do You List the Vertices, Edges, and Faces of a Polyhedron? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-4: Additional Example 3 with Try Another One Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Example 3 with Try Another One Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Topic 13: Assessment Form A (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Identify space figures and their relationships with polygons to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Topic 13: Assessment Form A Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Identify space figures and their relationships with polygons to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Topic 13: Assessment Form B Topic 13: Assessment Form B (PDF) Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Identify space figures and their relationships with polygons to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. 13-3: Ex 4: Apply the Volumes of Cones to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: MathXL for School: Enrichment Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: MathXL for School: Enrichment Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-1: Ex 3: Describe a Cross Section & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-4: MathXL for School: Reteach to Build Understanding Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Ex 3: Apply the Volumes of Pyramids to Solve Problems & Try It! Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 2: Use the Volumes of Spheres to Solve Problems & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: Ex 3: Find the Volumes of Hemispheres & Try It! Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-4: MathXL for School: Enrichment Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Ex 5: Rotate a Polygon to Form a Three-Dimensional Figure & Try It! Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: MathXL for School: Reteach to Build Understanding Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Virtual Nerd™: What is a Solid? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: Virtual Nerd™: How Do You Find the Volume of a Composite Figure? Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: MathXL for School: Reteach to Build Understanding Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-3: Ex 1: Apply Cavalieri’s Principle to Pyramids and Cones & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-2: Ex 1: Develop Cavalieri’s Principle & Try It! Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. 13-3: Virtual Nerd™: What is the Formula for the Volume of a Cone? Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-1: Virtual Nerd™: How Do You List the Vertices, Edges, and Faces of a Polyhedron? Curriculum Standards: Identify space figures and their relationships with polygons to solve problems. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 13-4: Additional Example 3 with Try Another One Curriculum Standards: Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-2: MathXL for School: Enrichment Curriculum Standards: Use the properties of prisms and cylinders to calculate their volumes. Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 13-3: Additional Example 3 with Try Another One Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Topic 13: Assessment Form C Curriculum Standards: Use the volumes of right and oblique pyramids and cones to solve problems. Calculate the volume of a sphere and solve problems involving the volumes of spheres. Use the properties of prisms and cylinders to calculate their volumes. Identify space figures and their relationships with polygons to solve problems. The student will use surface area and volume of three-dimensional objects to solve practical problems. determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. End-of-Course Assessment (PDF) End-of-Course Assessment Curriculum Standards: Use the equations and graphs of circles to solve problems. Write, graph, and apply the equation of a circle. Use similarity and the geometric mean to solve problems involving right triangles. Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. Apply theorems about isosceles and equilateral triangles to solve problems. Determine congruent triangles by comparing two angles and one side. Identify congruent right triangles. Use triangle congruence to solve problems with overlapping triangles. Use the relationships between sides, segments, and angles of triangles to solve problems. Find the sums of the measures of the exterior angles and interior angles of polygons. Use triangle congruence to understand kites and trapezoids. Use the properties of rhombuses, rectangles, and squares to solve problems. Graph quadratic functions using standard form. Write and graph quadratic functions in standard form. Use trigonometric ratios to find lengths and angle measures of right triangles. Use trigonometry to solve problems. Reason about operations with real numbers. Dilate figures and identify characteristics of dilations. Determine whether figures are similar. Solve problems using the measures of interior and exterior angles of triangles. Identify and describe transformations of two-dimensional figures. Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. Use theorems to compare the sides and angles of a triangle. Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. Use the Law of Sines to solve problems. Draw and describe the reflection of a figure across a line of reflection. Describe the properties of a figure before and after translation. Use dilation and rigid motion to establish triangle similarity theorems. Write conditionals and biconditionals and find their truth values. Use SAS and SSS to determine whether triangles are congruent. Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. Use the midpoint and distance formulas to solve problems. Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. Identify, evaluate, and graph linear functions. Determine the measures of the angles formed when parallel lines are intersected by a transversal. Rewrite and use literal equations to solve problems. Relate the length of a chord to the central angle it subtends and the arc it intercepts. Use relationships between circles, angles, and arcs. Write equations of parallel lines and perpendicular lines. Use slope to solve problems about parallel and perpendicular lines. Use the equations and graphs of parabolas to solve problems. Understand and apply the geometric properties of a parabola. Use the Law of Cosines to solve problems. Use the Law of Sines and the Law of Cosines. Use properties of segments and angles to find their measures. The student will solve problems involving equations of circles. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. Know and apply properties of a circle to solve problems and logically justify results. Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. The student, given information in the form of a figure or statement, will prove two triangles are similar. the Pythagorean Theorem and its converse; comparing ratios between lengths, perimeters, areas, and volumes of similar figures; Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. investigating and using formulas for determining distance, midpoint, and slope; the perpendicular bisector of a line segment; ordering the sides by length, given angle measures; ordering the angles by degree measure, given side lengths; determining whether a triangle exists; and determining the range in which the length of the third side must lie. The student, given information in the form of a figure or statement, will prove two triangles are congruent. The student will verify and use properties of quadrilaterals to solve problems, including practical problems. sum of the interior and/or exterior angles; measure of an interior and/or exterior angle; and number of sides of a regular polygon. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Know and apply properties of congruent and similar figures to solve problems and logically justify results. trigonometric ratios. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. solving problems, including practical problems, about similar geometric figures. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. (HONORS ONLY) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle) Use numeric, graphic and algebraic representations of transformations in two dimensions, such as reflections, translations, dilations, and rotations about the origin by multiples of 90?, to solve problems involving figures on a coordinate plane and identify types of symmetry. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. identifying the converse, inverse, and contrapositive of a conditional statement; translating a short verbal argument into symbolic form; and determining the validity of a logical argument. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Assess the validity of a logical argument and give counterexamples to disprove a statement. Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c properties of special right triangles; and Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. prove two or more lines are parallel; and solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. angle measures formed by intersecting chords, secants, and/or tangents; lengths of segments formed by intersecting chords, secants, and/or tangents; arc length; and Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. applying slope to verify and determine whether lines are parallel or perpendicular; Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. Prove the Laws of Sines and Cosines and use them to solve problems. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. (HONORS ONLY) Prove the Laws of Sines and Cosines and use them to solve problems. a line segment congruent to a given line segment; an angle congruent to a given angle; Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Mathematics II Next-Generation Practice Test Practice Performance Tasks 3/4-Year Practice Performance Task 1 3/4-Year Practice Performance Task 2 Credits, enVision Integrated Mathematics II Teacher Resources Container Teacher's Edition Program Overview Intended Role: Instructor Answers & Solutions Intended Role: Instructor ExamView® Download (Windows) Intended Role: Instructor ExamView® Download (Macintosh) Intended Role: Instructor Multilingual Handbook Download (zip) Intended Role: Instructor Teacher's Edition eText: Mathematics II Intended Role: Instructor Step 1: Explore Intended Role: Instructor Step 2: Understand & Apply Intended Role: Instructor Step 3: Practice & Problem Solving Intended Role: Instructor Step 4: Assess & Differentiate Intended Role: Instructor enVision STEM Project Intended Role: Instructor Mathematical Modeling in 3 Acts Intended Role: Instructor Beginning-of-Year Assessment (Editable) Intended Role: Instructor Beginning-of-Year Assessment: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 1 Intended Role: Instructor Topic 1: Readiness Assessment (Editable) Intended Role: Instructor Topic 1: Readiness Assessment: Answer Key Intended Role: Instructor Topic 1: enVision STEM Masters: Answer Key Intended Role: Instructor 1-1: Lesson Plan (PDF) Intended Role: Instructor 1-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 1-1 Intended Role: Instructor 1-1: Lesson Quiz (Editable) Intended Role: Instructor 1-1: Lesson Quiz: Answer Key Intended Role: Instructor 1-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 1-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 1-1: Additional Practice (Editable) Intended Role: Instructor 1-1: Additional Practice: Answer Key Intended Role: Instructor 1-1: Enrichment (Editable) Intended Role: Instructor 1-1: Enrichment: Answer Key Intended Role: Instructor 1-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 1-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 1-2: Lesson Plan (PDF) Intended Role: Instructor 1-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 1-2 Intended Role: Instructor 1-2: Lesson Quiz (Editable) Intended Role: Instructor 1-2: Lesson Quiz: Answer Key Intended Role: Instructor 1-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 1-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 1-2: Additional Practice (Editable) Intended Role: Instructor 1-2: Additional Practice: Answer Key Intended Role: Instructor 1-2: Enrichment (Editable) Intended Role: Instructor 1-2: Enrichment: Answer Key Intended Role: Instructor 1-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 1-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 1-3: Lesson Plan (PDF) Intended Role: Instructor 1-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 1-3 Intended Role: Instructor 1-3: Lesson Quiz (Editable) Intended Role: Instructor 1-3: Lesson Quiz: Answer Key Intended Role: Instructor 1-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 1-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 1-3: Additional Practice (Editable) Intended Role: Instructor 1-3: Additional Practice: Answer Key Intended Role: Instructor 1-3: Enrichment (Editable) Intended Role: Instructor 1-3: Enrichment: Answer Key Intended Role: Instructor 1-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 1-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 1-4: Lesson Plan (PDF) Intended Role: Instructor 1-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 1-4 Intended Role: Instructor 1-4: Lesson Quiz (Editable) Intended Role: Instructor 1-4: Lesson Quiz: Answer Key Intended Role: Instructor 1-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 1-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 1-4: Additional Practice (Editable) Intended Role: Instructor 1-4: Additional Practice: Answer Key Intended Role: Instructor 1-4: Enrichment (Editable) Intended Role: Instructor 1-4: Enrichment: Answer Key Intended Role: Instructor 1-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 1-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 1-5: Lesson Plan (PDF) Intended Role: Instructor 1-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 1-5 Intended Role: Instructor 1-5: Lesson Quiz (Editable) Intended Role: Instructor 1-5: Lesson Quiz: Answer Key Intended Role: Instructor 1-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 1-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 1-5: Additional Practice (Editable) Intended Role: Instructor 1-5: Additional Practice: Answer Key Intended Role: Instructor 1-5: Enrichment (Editable) Intended Role: Instructor 1-5: Enrichment: Answer Key Intended Role: Instructor 1-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 1-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 1: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 1: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 1: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 1: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 1: Assessment Form A (Editable) Intended Role: Instructor Topic 1: Assessment Form A: Answer Key Intended Role: Instructor Topic 1: Assessment Form B (Editable) Intended Role: Instructor Topic 1: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 2 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 2 Intended Role: Instructor Topic 2: Readiness Assessment (Editable) Intended Role: Instructor Topic 2: Readiness Assessment: Answer Key Intended Role: Instructor Topic 2: enVision STEM Masters: Answer Key Intended Role: Instructor 2-1: Lesson Plan (PDF) Intended Role: Instructor 2-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-1 Intended Role: Instructor 2-1: Lesson Quiz (Editable) Intended Role: Instructor 2-1: Lesson Quiz: Answer Key Intended Role: Instructor 2-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-1: Additional Practice (Editable) Intended Role: Instructor 2-1: Additional Practice: Answer Key Intended Role: Instructor 2-1: Enrichment (Editable) Intended Role: Instructor 2-1: Enrichment: Answer Key Intended Role: Instructor 2-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-2: Lesson Plan (PDF) Intended Role: Instructor 2-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-2 Intended Role: Instructor 2-2: Lesson Quiz (Editable) Intended Role: Instructor 2-2: Lesson Quiz: Answer Key Intended Role: Instructor 2-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-2: Additional Practice (Editable) Intended Role: Instructor 2-2: Additional Practice: Answer Key Intended Role: Instructor 2-2: Enrichment (Editable) Intended Role: Instructor 2-2: Enrichment: Answer Key Intended Role: Instructor 2-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-3: Lesson Plan (PDF) Intended Role: Instructor 2-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-3 Intended Role: Instructor 2-3: Lesson Quiz (Editable) Intended Role: Instructor 2-3: Lesson Quiz: Answer Key Intended Role: Instructor 2-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-3: Additional Practice (Editable) Intended Role: Instructor 2-3: Additional Practice: Answer Key Intended Role: Instructor 2-3: Enrichment (Editable) Intended Role: Instructor 2-3: Enrichment: Answer Key Intended Role: Instructor 2-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-4: Lesson Plan (PDF) Intended Role: Instructor 2-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-4 Intended Role: Instructor 2-4: Lesson Quiz (Editable) Intended Role: Instructor 2-4: Lesson Quiz: Answer Key Intended Role: Instructor 2-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-4: Additional Practice (Editable) Intended Role: Instructor 2-4: Additional Practice: Answer Key Intended Role: Instructor 2-4: Enrichment (Editable) Intended Role: Instructor 2-4: Enrichment: Answer Key Intended Role: Instructor 2-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-5: Lesson Plan (PDF) Intended Role: Instructor 2-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-5 Intended Role: Instructor 2-5: Lesson Quiz (Editable) Intended Role: Instructor 2-5: Lesson Quiz: Answer Key Intended Role: Instructor 2-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-5: Additional Practice (Editable) Intended Role: Instructor 2-5: Additional Practice: Answer Key Intended Role: Instructor 2-5: Enrichment (Editable) Intended Role: Instructor 2-5: Enrichment: Answer Key Intended Role: Instructor 2-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-6: Lesson Plan (PDF) Intended Role: Instructor 2-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-6 Intended Role: Instructor 2-6: Lesson Quiz (Editable) Intended Role: Instructor 2-6: Lesson Quiz: Answer Key Intended Role: Instructor 2-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-6: Additional Practice (Editable) Intended Role: Instructor 2-6: Additional Practice: Answer Key Intended Role: Instructor 2-6: Enrichment (Editable) Intended Role: Instructor 2-6: Enrichment: Answer Key Intended Role: Instructor 2-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 2-7: Lesson Plan (PDF) Intended Role: Instructor 2-7: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 2-7 Intended Role: Instructor 2-7: Lesson Quiz (Editable) Intended Role: Instructor 2-7: Lesson Quiz: Answer Key Intended Role: Instructor 2-7: Reteach to Build Understanding (Editable) Intended Role: Instructor 2-7: Reteach to Build Understanding: Answer Key Intended Role: Instructor 2-7: Additional Practice (Editable) Intended Role: Instructor 2-7: Additional Practice: Answer Key Intended Role: Instructor 2-7: Enrichment (Editable) Intended Role: Instructor 2-7: Enrichment: Answer Key Intended Role: Instructor 2-7: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 2-7: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 2: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 2: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 2: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 2: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 2: Assessment Form A (Editable) Intended Role: Instructor Topic 2: Assessment Form A: Answer Key Intended Role: Instructor Topic 2: Assessment Form B (Editable) Intended Role: Instructor Topic 2: Assessment Form B: Answer Key Intended Role: Instructor Benchmark Test 1 (Editable) Intended Role: Instructor Benchmark Test 1: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 3 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 3 Intended Role: Instructor Topic 3: Readiness Assessment (Editable) Intended Role: Instructor Topic 3: Readiness Assessment: Answer Key Intended Role: Instructor Topic 3: enVision STEM Masters: Answer Key Intended Role: Instructor 3-1: Lesson Plan (PDF) Intended Role: Instructor 3-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 3-1 Intended Role: Instructor 3-1: Lesson Quiz (Editable) Intended Role: Instructor 3-1: Lesson Quiz: Answer Key Intended Role: Instructor 3-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 3-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 3-1: Additional Practice (Editable) Intended Role: Instructor 3-1: Additional Practice: Answer Key Intended Role: Instructor 3-1: Enrichment (Editable) Intended Role: Instructor 3-1: Enrichment: Answer Key Intended Role: Instructor 3-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 3-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 3-2: Lesson Plan (PDF) Intended Role: Instructor 3-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 3-2 Intended Role: Instructor 3-2: Lesson Quiz (Editable) Intended Role: Instructor 3-2: Lesson Quiz: Answer Key Intended Role: Instructor 3-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 3-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 3-2: Additional Practice (Editable) Intended Role: Instructor 3-2: Additional Practice: Answer Key Intended Role: Instructor 3-2: Enrichment (Editable) Intended Role: Instructor 3-2: Enrichment: Answer Key Intended Role: Instructor 3-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 3-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 3-3: Lesson Plan (PDF) Intended Role: Instructor 3-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 3-3 Intended Role: Instructor 3-3: Lesson Quiz (Editable) Intended Role: Instructor 3-3: Lesson Quiz: Answer Key Intended Role: Instructor 3-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 3-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 3-3: Additional Practice (Editable) Intended Role: Instructor 3-3: Additional Practice: Answer Key Intended Role: Instructor 3-3: Enrichment (Editable) Intended Role: Instructor 3-3: Enrichment: Answer Key Intended Role: Instructor 3-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 3-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 3-4: Lesson Plan (PDF) Intended Role: Instructor 3-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 3-4 Intended Role: Instructor 3-4: Lesson Quiz (Editable) Intended Role: Instructor 3-4: Lesson Quiz: Answer Key Intended Role: Instructor 3-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 3-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 3-4: Additional Practice (Editable) Intended Role: Instructor 3-4: Additional Practice: Answer Key Intended Role: Instructor 3-4: Enrichment (Editable) Intended Role: Instructor 3-4: Enrichment: Answer Key Intended Role: Instructor 3-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 3-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 3-5: Lesson Plan (PDF) Intended Role: Instructor 3-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 3-5 Intended Role: Instructor 3-5: Lesson Quiz (Editable) Intended Role: Instructor 3-5: Lesson Quiz: Answer Key Intended Role: Instructor 3-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 3-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 3-5: Additional Practice (Editable) Intended Role: Instructor 3-5: Additional Practice: Answer Key Intended Role: Instructor 3-5: Enrichment (Editable) Intended Role: Instructor 3-5: Enrichment: Answer Key Intended Role: Instructor 3-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 3-5:Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 3: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 3: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 3: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 3: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 3: Assessment Form A (Editable) Intended Role: Instructor Topic 3: Assessment Form A: Answer Key Intended Role: Instructor Topic 3: Assessment Form B (Editable) Intended Role: Instructor Topic 3: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 4 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 4 Intended Role: Instructor Topic 4: Readiness Assessment (Editable) Intended Role: Instructor Topic 4: Readiness Assessment: Answer Key Intended Role: Instructor Topic 4: enVision STEM Masters: Answer Key Intended Role: Instructor 4-1: Lesson Plan (PDF) Intended Role: Instructor 4-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 4-1 Intended Role: Instructor 4-1: Lesson Quiz (Editable) Intended Role: Instructor 4-1: Lesson Quiz: Answer Key Intended Role: Instructor 4-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 4-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 4-1: Additional Practice (Editable) Intended Role: Instructor 4-1: Additional Practice: Answer Key Intended Role: Instructor 4-1: Enrichment (Editable) Intended Role: Instructor 4-1: Enrichment: Answer Key Intended Role: Instructor 4-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 4-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 4-2: Lesson Plan (PDF) Intended Role: Instructor 4-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 4-2 Intended Role: Instructor 4-2: Lesson Quiz (Editable) Intended Role: Instructor 4-2: Lesson Quiz: Answer Key Intended Role: Instructor 4-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 4-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 4-2: Additional Practice (Editable) Intended Role: Instructor 4-2: Additional Practice: Answer Key Intended Role: Instructor 4-2: Enrichment (Editable) Intended Role: Instructor 4-2: Enrichment: Answer Key Intended Role: Instructor 4-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 4-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 4-3: Lesson Plan (PDF) Intended Role: Instructor 4-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 4-3 Intended Role: Instructor 4-3: Lesson Quiz (Editable) Intended Role: Instructor 4-3: Lesson Quiz: Answer Key Intended Role: Instructor 4-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 4-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 4-3: Additional Practice (Editable) Intended Role: Instructor 4-3: Additional Practice: Answer Key Intended Role: Instructor 4-3: Enrichment (Editable) Intended Role: Instructor 4-3: Enrichment: Answer Key Intended Role: Instructor 4-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 4-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 4-4: Lesson Plan (PDF) Intended Role: Instructor 4-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 4-4 Intended Role: Instructor 4-4: Lesson Quiz (Editable) Intended Role: Instructor 4-4: Lesson Quiz: Answer Key Intended Role: Instructor 4-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 4-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 4-4: Additional Practice (Editable) Intended Role: Instructor 4-4: Additional Practice: Answer Key Intended Role: Instructor 4-4: Enrichment (Editable) Intended Role: Instructor 4-4: Enrichment: Answer Key Intended Role: Instructor 4-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 4-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 4-5: Lesson Plan (PDF) Intended Role: Instructor 4-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 4-5 Intended Role: Instructor 4-5: Lesson Quiz (Editable) Intended Role: Instructor 4-5: Lesson Quiz: Answer Key Intended Role: Instructor 4-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 4-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 4-5: Additional Practice (Editable) Intended Role: Instructor 4-5: Additional Practice: Answer Key Intended Role: Instructor 4-5: Enrichment (Editable) Intended Role: Instructor 4-5: Enrichment: Answer Key Intended Role: Instructor 4-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 4-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 4: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 4: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 4: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 4: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 4: Assessment Form A (Editable) Intended Role: Instructor Topic 4: Assessment Form A: Answer Key Intended Role: Instructor Topic 4: Assessment Form B (Editable) Intended Role: Instructor Topic 4: Assessment Form B: Answer Key Intended Role: Instructor Benchmark Test 2 (Editable) Intended Role: Instructor Benchmark Test 2: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 5 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 5 Intended Role: Instructor Topic 5: Readiness Assessment (Editable) Intended Role: Instructor Topic 5: Readiness Assessment: Answer Key Intended Role: Instructor Topic 5: enVision STEM Masters: Answer Key Intended Role: Instructor 5-1: Lesson Plan (PDF) Intended Role: Instructor 5-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 5-1 Intended Role: Instructor 5-1: Lesson Quiz (Editable) Intended Role: Instructor 5-1: Lesson Quiz: Answer Key Intended Role: Instructor 5-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 5-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 5-1: Additional Practice (Editable) Intended Role: Instructor 5-1: Additional Practice: Answer Key Intended Role: Instructor 5-1: Enrichment (Editable) Intended Role: Instructor 5-1: Enrichment: Answer Key Intended Role: Instructor 5-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 5-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 5-2: Lesson Plan (PDF) Intended Role: Instructor 5-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 5-2 Intended Role: Instructor 5-2: Lesson Quiz (Editable) Intended Role: Instructor 5-2: Lesson Quiz: Answer Key Intended Role: Instructor 5-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 5-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 5-2: Additional Practice (Editable) Intended Role: Instructor 5-2: Additional Practice: Answer Key Intended Role: Instructor 5-2: Enrichment (Editable) Intended Role: Instructor 5-2: Enrichment: Answer Key Intended Role: Instructor 5-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 5-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 5-3: Lesson Plan (PDF) Intended Role: Instructor 5-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 5-3 Intended Role: Instructor 5-3: Lesson Quiz (Editable) Intended Role: Instructor 5-3: Lesson Quiz: Answer Key Intended Role: Instructor 5-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 5-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 5-3: Additional Practice (Editable) Intended Role: Instructor 5-3: Additional Practice: Answer Key Intended Role: Instructor 5-3: Enrichment (Editable) Intended Role: Instructor 5-3: Enrichment: Answer Key Intended Role: Instructor 5-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 5-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 5-4: Lesson Plan (PDF) Intended Role: Instructor 5-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 5-4 Intended Role: Instructor 5-4: Lesson Quiz (Editable) Intended Role: Instructor 5-4: Lesson Quiz: Answer Key Intended Role: Instructor 5-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 5-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 5-4: Additional Practice (Editable) Intended Role: Instructor 5-4: Additional Practice: Answer Key Intended Role: Instructor 5-4: Enrichment (Editable) Intended Role: Instructor 5-4: Enrichment: Answer Key Intended Role: Instructor 5-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 5-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 5-5: Lesson Plan (PDF) Intended Role: Instructor 5-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 5-5 Intended Role: Instructor 5-5: Lesson Quiz (Editable) Intended Role: Instructor 5-5: Lesson Quiz: Answer Key Intended Role: Instructor 5-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 5-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 5-5: Additional Practice (Editable) Intended Role: Instructor 5-5: Additional Practice: Answer Key Intended Role: Instructor 5-5: Enrichment (Editable) Intended Role: Instructor 5-5: Enrichment: Answer Key Intended Role: Instructor 5-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 5-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Teacher's Edition eText: Lesson 5-5a Intended Role: Instructor Topic 5: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 5: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 5: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 5: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 5: Assessment Form A (Editable) Intended Role: Instructor Topic 5: Assessment Form A: Answer Key Intended Role: Instructor Topic 5: Assessment Form B (Editable) Intended Role: Instructor Topic 5: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 6 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 6 Intended Role: Instructor Topic 6: Readiness Assessment (Editable) Intended Role: Instructor Topic 6: Readiness Assessment: Answer Key Intended Role: Instructor Topic 6: enVision STEM Masters: Answer Key Intended Role: Instructor 6-1: Lesson Plan (PDF) Intended Role: Instructor 6-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-1 Intended Role: Instructor 6-1: Lesson Quiz (Editable) Intended Role: Instructor 6-1: Lesson Quiz: Answer Key Intended Role: Instructor 6-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-1: Additional Practice (Editable) Intended Role: Instructor 6-1: Additional Practice: Answer Key Intended Role: Instructor 6-1: Enrichment (Editable) Intended Role: Instructor 6-1: Enrichment: Answer Key Intended Role: Instructor 6-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-2: Lesson Plan (PDF) Intended Role: Instructor 6-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-2 Intended Role: Instructor 6-2: Lesson Quiz (Editable) Intended Role: Instructor 6-2: Lesson Quiz: Answer Key Intended Role: Instructor 6-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-2: Additional Practice (Editable) Intended Role: Instructor 6-2: Additional Practice: Answer Key Intended Role: Instructor 6-2: Enrichment (Editable) Intended Role: Instructor 6-2: Enrichment: Answer Key Intended Role: Instructor 6-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-3: Lesson Plan (PDF) Intended Role: Instructor 6-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-3 Intended Role: Instructor 6-3: Lesson Quiz (Editable) Intended Role: Instructor 6-3: Lesson Quiz: Answer Key Intended Role: Instructor 6-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-3: Additional Practice (Editable) Intended Role: Instructor 6-3: Additional Practice: Answer Key Intended Role: Instructor 6-3: Enrichment (Editable) Intended Role: Instructor 6-3: Enrichment: Answer Key Intended Role: Instructor 6-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-4: Lesson Plan (PDF) Intended Role: Instructor 6-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-4 Intended Role: Instructor 6-4: Lesson Quiz (Editable) Intended Role: Instructor 6-4: Lesson Quiz: Answer Key Intended Role: Instructor 6-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-4: Additional Practice (Editable) Intended Role: Instructor 6-4: Additional Practice: Answer Key Intended Role: Instructor 6-4: Enrichment (Editable) Intended Role: Instructor 6-4: Enrichment: Answer Key Intended Role: Instructor 6-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-5: Lesson Plan (PDF) Intended Role: Instructor 6-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-5 Intended Role: Instructor 6-5: Lesson Quiz (Editable) Intended Role: Instructor 6-5: Lesson Quiz: Answer Key Intended Role: Instructor 6-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-5: Additional Practice (Editable) Intended Role: Instructor 6-5: Additional Practice: Answer Key Intended Role: Instructor 6-5: Enrichment (Editable) Intended Role: Instructor 6-5: Enrichment: Answer Key Intended Role: Instructor 6-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-6: Lesson Plan (PDF) Intended Role: Instructor 6-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-6 Intended Role: Instructor 6-6: Lesson Quiz (Editable) Intended Role: Instructor 6-6: Lesson Quiz: Answer Key Intended Role: Instructor 6-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-6: Additional Practice (Editable) Intended Role: Instructor 6-6: Additional Practice: Answer Key Intended Role: Instructor 6-6: Enrichment (Editable) Intended Role: Instructor 6-6: Enrichment: Answer Key Intended Role: Instructor 6-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-7: Lesson Plan (PDF) Intended Role: Instructor 6-7: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-7 Intended Role: Instructor 6-7: Lesson Quiz (Editable) Intended Role: Instructor 6-7: Lesson Quiz: Answer Key Intended Role: Instructor 6-7: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-7: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-7: Additional Practice (Editable) Intended Role: Instructor 6-7: Additional Practice: Answer Key Intended Role: Instructor 6-7: Enrichment (Editable) Intended Role: Instructor 6-7: Enrichment: Answer Key Intended Role: Instructor 6-7: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-7: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-8: Lesson Plan (PDF) Intended Role: Instructor 6-8: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-8 Intended Role: Instructor 6-8: Lesson Quiz (Editable) Intended Role: Instructor 6-8: Lesson Quiz: Answer Key Intended Role: Instructor 6-8: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-8: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-8: Additional Practice (Editable) Intended Role: Instructor 6-8: Additional Practice: Answer Key Intended Role: Instructor 6-8: Enrichment (Editable) Intended Role: Instructor 6-8: Enrichment: Answer Key Intended Role: Instructor 6-8: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-8: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 6-9: Lesson Plan (PDF) Intended Role: Instructor 6-9: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 6-9 Intended Role: Instructor 6-9: Lesson Quiz (Editable) Intended Role: Instructor 6-9: Lesson Quiz: Answer Key Intended Role: Instructor 6-9: Reteach to Build Understanding (Editable) Intended Role: Instructor 6-9: Reteach to Build Understanding: Answer Key Intended Role: Instructor 6-9: Additional Practice (Editable) Intended Role: Instructor 6-9: Additional Practice: Answer Key Intended Role: Instructor 6-9: Enrichment (Editable) Intended Role: Instructor 6-9: Enrichment: Answer Key Intended Role: Instructor 6-9: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 6-9: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 6: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 6: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 6: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 6: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 6: Assessment Form A (Editable) Intended Role: Instructor Topic 6: Assessment Form A: Answer Key Intended Role: Instructor Topic 6: Assessment Form B (Editable) Intended Role: Instructor Topic 6: Assessment Form B: Answer Key Intended Role: Instructor Mid-Year Assessment (Editable) Intended Role: Instructor Mid-Year Assessment: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 7 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 7 Intended Role: Instructor Topic 7: Readiness Assessment (Editable) Intended Role: Instructor Topic 7: Readiness Assessment: Answer Key Intended Role: Instructor Topic 7: enVision STEM Masters: Answer Key Intended Role: Instructor 7-1: Lesson Plan (PDF) Intended Role: Instructor 7-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-1 Intended Role: Instructor 7-1: Lesson Quiz (Editable) Intended Role: Instructor 7-1: Lesson Quiz: Answer Key Intended Role: Instructor 7-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-1: Additional Practice (Editable) Intended Role: Instructor 7-1: Additional Practice: Answer Key Intended Role: Instructor 7-1: Enrichment (Editable) Intended Role: Instructor 7-1: Enrichment: Answer Key Intended Role: Instructor 7-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-2: Lesson Plan (PDF) Intended Role: Instructor 7-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-2 Intended Role: Instructor 7-2: Lesson Quiz (Editable) Intended Role: Instructor 7-2: Lesson Quiz: Answer Key Intended Role: Instructor 7-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-2: Additional Practice (Editable) Intended Role: Instructor 7-2: Additional Practice: Answer Key Intended Role: Instructor 7-2: Enrichment (Editable) Intended Role: Instructor 7-2: Enrichment: Answer Key Intended Role: Instructor 7-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-3: Lesson Plan (PDF) Intended Role: Instructor 7-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-3 Intended Role: Instructor 7-3: Lesson Quiz (Editable) Intended Role: Instructor 7-3: Lesson Quiz: Answer Key Intended Role: Instructor 7-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-3: Additional Practice (Editable) Intended Role: Instructor 7-3: Additional Practice: Answer Key Intended Role: Instructor 7-3: Enrichment (Editable) Intended Role: Instructor 7-3: Enrichment: Answer Key Intended Role: Instructor 7-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-4: Lesson Plan (PDF) Intended Role: Instructor 7-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-4 Intended Role: Instructor 7-4: Lesson Quiz (Editable) Intended Role: Instructor 7-4: Lesson Quiz: Answer Key Intended Role: Instructor 7-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-4: Additional Practice (Editable) Intended Role: Instructor 7-4: Additional Practice: Answer Key Intended Role: Instructor 7-4: Enrichment (Editable) Intended Role: Instructor 7-4: Enrichment: Answer Key Intended Role: Instructor 7-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-5: Lesson Plan (PDF) Intended Role: Instructor 7-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-5 Intended Role: Instructor 7-5: Lesson Quiz (Editable) Intended Role: Instructor 7-5: Lesson Quiz: Answer Key Intended Role: Instructor 7-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-5: Additional Practice (Editable) Intended Role: Instructor 7-5: Additional Practice: Answer Key Intended Role: Instructor 7-5: Enrichment (Editable) Intended Role: Instructor 7-5: Enrichment: Answer Key Intended Role: Instructor 7-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-6: Lesson Plan (PDF) Intended Role: Instructor 7-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-6 Intended Role: Instructor 7-6: Lesson Quiz (Editable) Intended Role: Instructor 7-6: Lesson Quiz: Answer Key Intended Role: Instructor 7-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-6: Additional Practice (Editable) Intended Role: Instructor 7-6: Additional Practice: Answer Key Intended Role: Instructor 7-6: Enrichment (Editable) Intended Role: Instructor 7-6: Enrichment: Answer Key Intended Role: Instructor 7-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 7-7: Lesson Plan (PDF) Intended Role: Instructor 7-7: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 7-7 Intended Role: Instructor 7-7: Lesson Quiz (Editable) Intended Role: Instructor 7-7: Lesson Quiz: Answer Key Intended Role: Instructor 7-7: Reteach to Build Understanding (Editable) Intended Role: Instructor 7-7: Reteach to Build Understanding: Answer Key Intended Role: Instructor 7-7: Additional Practice (Editable) Intended Role: Instructor 7-7: Additional Practice: Answer Key Intended Role: Instructor 7-7: Enrichment (Editable) Intended Role: Instructor 7-7: Enrichment: Answer Key Intended Role: Instructor 7-7: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 7-7: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 7: Performance Assessment: Form A (Editable) Intended Role: Instructor Topic 7: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 7: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 7: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 7: Assessment: Form A: Answer Key Intended Role: Instructor Topic 7: Assessment: Form A (Editable) Intended Role: Instructor Topic 7: Assessment Form B (Editable) Intended Role: Instructor Topic 7: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 8 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 8 Intended Role: Instructor Topic 8: Readiness Assessment (Editable) Intended Role: Instructor Topic 8: Readiness Assessment: Answer Key Intended Role: Instructor Topic 8: enVision STEM Masters: Answer Key Intended Role: Instructor 8-1: Lesson Plan (PDF) Intended Role: Instructor 8-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-1 Intended Role: Instructor 8-1: Lesson Quiz (Editable) Intended Role: Instructor 8-1: Lesson Quiz: Answer Key Intended Role: Instructor 8-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-1: Additional Practice (Editable) Intended Role: Instructor 8-1: Additional Practice: Answer Key Intended Role: Instructor 8-1: Enrichment (Editable) Intended Role: Instructor 8-1: Enrichment: Answer Key Intended Role: Instructor 8-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 8-2: Lesson Plan (PDF) Intended Role: Instructor 8-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-2 Intended Role: Instructor 8-2: Lesson Quiz (Editable) Intended Role: Instructor 8-2: Lesson Quiz: Answer Key Intended Role: Instructor 8-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-2: Additional Practice (Editable) Intended Role: Instructor 8-2: Additional Practice: Answer Key Intended Role: Instructor 8-2: Enrichment (Editable) Intended Role: Instructor 8-2: Enrichment: Answer Key Intended Role: Instructor 8-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 8-3: Lesson Plan (PDF) Intended Role: Instructor 8-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-3 Intended Role: Instructor 8-3: Lesson Quiz (Editable) Intended Role: Instructor 8-3: Lesson Quiz: Answer Key Intended Role: Instructor 8-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-3: Additional Practice (Editable) Intended Role: Instructor 8-3: Additional Practice: Answer Key Intended Role: Instructor 8-3: Enrichment (Editable) Intended Role: Instructor 8-3: Enrichment: Answer Key Intended Role: Instructor 8-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 8-4: Lesson Plan (PDF) Intended Role: Instructor 8-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-4 Intended Role: Instructor 8-4: Lesson Quiz (Editable) Intended Role: Instructor 8-4: Lesson Quiz: Answer Key Intended Role: Instructor 8-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-4: Additional Practice (Editable) Intended Role: Instructor 8-4: Additional Practice: Answer Key Intended Role: Instructor 8-4: Enrichment (Editable) Intended Role: Instructor 8-4: Enrichment: Answer Key Intended Role: Instructor 8-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 8-5: Lesson Plan (PDF) Intended Role: Instructor 8-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-5 Intended Role: Instructor 8-5: Lesson Quiz (Editable) Intended Role: Instructor 8-5: Lesson Quiz: Answer Key Intended Role: Instructor 8-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-5: Additional Practice (Editable) Intended Role: Instructor 8-5: Additional Practice: Answer Key Intended Role: Instructor 8-5: Enrichment (Editable) Intended Role: Instructor 8-5: Enrichment: Answer Key Intended Role: Instructor 8-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 8-6: Lesson Plan (PDF) Intended Role: Instructor 8-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 8-6 Intended Role: Instructor 8-6: Lesson Quiz (Editable) Intended Role: Instructor 8-6: Lesson Quiz: Answer Key Intended Role: Instructor 8-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 8-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 8-6: Additional Practice (Editable) Intended Role: Instructor 8-6: Additional Practice: Answer Key Intended Role: Instructor 8-6: Enrichment (Editable) Intended Role: Instructor 8-6: Enrichment: Answer Key Intended Role: Instructor 8-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 8-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 8: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 8: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 8: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 8: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 8: Assessment Form A (Editable) Intended Role: Instructor Topic 8: Assessment Form A: Answer Key Intended Role: Instructor Topic 8: Assessment Form B (Editable) Intended Role: Instructor Topic 8: Assessment Form B: Answer Key Intended Role: Instructor Benchmark Test 3 (Editable) Intended Role: Instructor Benchmark Test 3: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 9 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 9 Intended Role: Instructor Topic 9: Readiness Assessment (Editable) Intended Role: Instructor Topic 9: Readiness Assessment: Answer Key Intended Role: Instructor Topic 9: enVision STEM Masters: Answer Key Intended Role: Instructor 9-1: Lesson Plan (PDF) Intended Role: Instructor 9-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-1 Intended Role: Instructor 9-1: Lesson Quiz (Editable) Intended Role: Instructor 9-1: Lesson Quiz: Answer Key Intended Role: Instructor 9-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-1: Additional Practice (Editable) Intended Role: Instructor 9-1: Additional Practice: Answer Key Intended Role: Instructor 9-1: Enrichment (Editable) Intended Role: Instructor 9-1: Enrichment: Answer Key Intended Role: Instructor 9-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-2: Lesson Plan (PDF) Intended Role: Instructor 9-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-2 Intended Role: Instructor 9-2: Lesson Quiz (Editable) Intended Role: Instructor 9-2: Lesson Quiz: Answer Key Intended Role: Instructor 9-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-2: Additional Practice (Editable) Intended Role: Instructor 9-2: Additional Practice: Answer Key Intended Role: Instructor 9-2: Enrichment (Editable) Intended Role: Instructor 9-2: Enrichment: Answer Key Intended Role: Instructor 9-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-3: Lesson Plan (PDF) Intended Role: Instructor 9-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-3 Intended Role: Instructor 9-3: Lesson Quiz (Editable) Intended Role: Instructor 9-3: Lesson Quiz: Answer Key Intended Role: Instructor 9-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-3: Additional Practice (Editable) Intended Role: Instructor 9-3: Additional Practice: Answer Key Intended Role: Instructor 9-3: Enrichment (Editable) Intended Role: Instructor 9-3: Enrichment: Answer Key Intended Role: Instructor 9-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-4: Lesson Plan (PDF) Intended Role: Instructor 9-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-4 Intended Role: Instructor 9-4: Lesson Quiz (Editable) Intended Role: Instructor 9-4: Lesson Quiz: Answer Key Intended Role: Instructor 9-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-4: Additional Practice (Editable) Intended Role: Instructor 9-4: Additional Practice: Answer Key Intended Role: Instructor 9-4: Enrichment (Editable) Intended Role: Instructor 9-4: Enrichment: Answer Key Intended Role: Instructor 9-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-5: Lesson Plan (PDF) Intended Role: Instructor 9-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-5 Intended Role: Instructor 9-5: Lesson Quiz (Editable) Intended Role: Instructor 9-5: Lesson Quiz: Answer Key Intended Role: Instructor 9-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-5: Additional Practice (Editable) Intended Role: Instructor 9-5: Additional Practice: Answer Key Intended Role: Instructor 9-5: Enrichment (Editable) Intended Role: Instructor 9-5: Enrichment: Answer Key Intended Role: Instructor 9-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-6: Lesson Plan (PDF) Intended Role: Instructor 9-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-6 Intended Role: Instructor 9-6: Lesson Quiz (Editable) Intended Role: Instructor 9-6: Lesson Quiz: Answer Key Intended Role: Instructor 9-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-6: Additional Practice (Editable) Intended Role: Instructor 9-6: Additional Practice: Answer Key Intended Role: Instructor 9-6: Enrichment (Editable) Intended Role: Instructor 9-6: Enrichment: Answer Key Intended Role: Instructor 9-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 9-7: Lesson Plan (PDF) Intended Role: Instructor 9-7: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 9-7 Intended Role: Instructor 9-7: Lesson Quiz (Editable) Intended Role: Instructor 9-7: Lesson Quiz: Answer Key Intended Role: Instructor 9-7: Reteach to Build Understanding (Editable) Intended Role: Instructor 9-7: Reteach to Build Understanding: Answer Key Intended Role: Instructor 9-7: Additional Practice (Editable) Intended Role: Instructor 9-7: Additional Practice: Answer Key Intended Role: Instructor 9-7: Enrichment (Editable) Intended Role: Instructor 9-7: Enrichment: Answer Key Intended Role: Instructor 9-7: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 9-7: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Teacher's Edition eText: Lesson 9-7a Intended Role: Instructor Topic 9: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 9: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 9: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 9: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 9: Assessment Form A (Editable) Intended Role: Instructor Topic 9: Assessment Form A: Answer Key Intended Role: Instructor Topic 9: Assessment Form B (Editable) Intended Role: Instructor Topic 9: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 10 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 10 Intended Role: Instructor Topic 10: Readiness Assessment (Editable) Intended Role: Instructor Topic 10: Readiness Assessment: Answer Key Intended Role: Instructor Topic 10: enVision STEM Masters: Answer Key Intended Role: Instructor 10-1: Lesson Plan (PDF) Intended Role: Instructor 10-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-1 Intended Role: Instructor 10-1: Lesson Quiz (Editable) Intended Role: Instructor 10-1: Lesson Quiz: Answer Key Intended Role: Instructor 10-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-1: Additional Practice (Editable) Intended Role: Instructor 10-1: Additional Practice: Answer Key Intended Role: Instructor 10-1: Enrichment (Editable) Intended Role: Instructor 10-1: Enrichment: Answer Key Intended Role: Instructor 10-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-2: Lesson Plan (PDF) Intended Role: Instructor 10-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-2 Intended Role: Instructor 10-2: Lesson Quiz (Editable) Intended Role: Instructor 10-2: Lesson Quiz: Answer Key Intended Role: Instructor 10-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-2: Additional Practice (Editable) Intended Role: Instructor 10-2: Additional Practice: Answer Key Intended Role: Instructor 10-2: Enrichment (Editable) Intended Role: Instructor 10-2: Enrichment: Answer Key Intended Role: Instructor 10-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-3: Lesson Plan (PDF) Intended Role: Instructor 10-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-3 Intended Role: Instructor 10-3: Lesson Quiz (Editable) Intended Role: Instructor 10-3: Lesson Quiz: Answer Key Intended Role: Instructor 10-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-3: Additional Practice (Editable) Intended Role: Instructor 10-3: Additional Practice: Answer Key Intended Role: Instructor 10-3: Enrichment (Editable) Intended Role: Instructor 10-3: Enrichment: Answer Key Intended Role: Instructor 10-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-4: Lesson Plan (PDF) Intended Role: Instructor 10-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-4 Intended Role: Instructor 10-4: Lesson Quiz (Editable) Intended Role: Instructor 10-4: Lesson Quiz: Answer Key Intended Role: Instructor 10-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-4: Additional Practice (Editable) Intended Role: Instructor 10-4: Additional Practice: Answer Key Intended Role: Instructor 10-4: Enrichment (Editable) Intended Role: Instructor 10-4: Enrichment: Answer Key Intended Role: Instructor 10-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-5: Lesson Plan (PDF) Intended Role: Instructor 10-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-5 Intended Role: Instructor 10-5: Lesson Quiz (Editable) Intended Role: Instructor 10-5: Lesson Quiz: Answer Key Intended Role: Instructor 10-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-5: Additional Practice (Editable) Intended Role: Instructor 10-5: Additional Practice: Answer Key Intended Role: Instructor 10-5: Enrichment (Editable) Intended Role: Instructor 10-5: Enrichment: Answer Key Intended Role: Instructor 10-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-6: Lesson Plan (PDF) Intended Role: Instructor 10-6: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-6 Intended Role: Instructor 10-6: Lesson Quiz (Editable) Intended Role: Instructor 10-6: Lesson Quiz: Answer Key Intended Role: Instructor 10-6: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-6: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-6: Additional Practice (Editable) Intended Role: Instructor 10-6: Additional Practice: Answer Key Intended Role: Instructor 10-6: Enrichment (Editable) Intended Role: Instructor 10-6: Enrichment: Answer Key Intended Role: Instructor 10-6: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-6: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 10-7: Lesson Plan (PDF) Intended Role: Instructor 10-7: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 10-7 Intended Role: Instructor 10-7: Lesson Quiz (Editable) Intended Role: Instructor 10-7: Lesson Quiz: Answer Key Intended Role: Instructor 10-7: Reteach to Build Understanding (Editable) Intended Role: Instructor 10-7: Reteach to Build Understanding: Answer Key Intended Role: Instructor 10-7: Additional Practice (Editable) Intended Role: Instructor 10-7: Additional Practice: Answer Key Intended Role: Instructor 10-7: Enrichment (Editable) Intended Role: Instructor 10-7: Enrichment: Answer Key Intended Role: Instructor 10-7: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 10-7: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 10: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 10: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 10: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 10: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 10: Assessment Form A (Editable) Intended Role: Instructor Topic 10: Assessment Form A: Answer Key Intended Role: Instructor Topic 10: Assessment Form B (Editable) Intended Role: Instructor Topic 10: Assessment Form B: Answer Key Intended Role: Instructor Benchmark Test 4 (Editable) Intended Role: Instructor Benchmark Test 4: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 11 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 11 Intended Role: Instructor Topic 11: Readiness Assessment (Editable) Intended Role: Instructor Topic 11: Readiness Assessment: Answer Key Intended Role: Instructor Topic 11: enVision STEM Masters: Answer Key Intended Role: Instructor 11-1: Lesson Plan (PDF) Intended Role: Instructor 11-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 11-1 Intended Role: Instructor 11-1: Lesson Quiz (Editable) Intended Role: Instructor 11-1: Lesson Quiz: Answer Key Intended Role: Instructor 11-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 11-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 11-1: Additional Practice (Editable) Intended Role: Instructor 11-1: Additional Practice: Answer Key Intended Role: Instructor 11-1: Enrichment (Editable) Intended Role: Instructor 11-1: Enrichment: Answer Key Intended Role: Instructor 11-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 11-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 11-2: Lesson Plan (PDF) Intended Role: Instructor 11-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 11-2 Intended Role: Instructor 11-2: Lesson Quiz (Editable) Intended Role: Instructor 11-2: Lesson Quiz: Answer Key Intended Role: Instructor 11-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 11-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 11-2: Additional Practice (Editable) Intended Role: Instructor 11-2: Additional Practice: Answer Key Intended Role: Instructor 11-2: Enrichment (Editable) Intended Role: Instructor 11-2: Enrichment: Answer Key Intended Role: Instructor 11-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 11-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 11-3: Lesson Plan (PDF) Intended Role: Instructor 11-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 11-3 Intended Role: Instructor 11-3: Lesson Quiz (Editable) Intended Role: Instructor 11-3: Lesson Quiz: Answer Key Intended Role: Instructor 11-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 11-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 11-3: Additional Practice (Editable) Intended Role: Instructor 11-3: Additional Practice: Answer Key Intended Role: Instructor 11-3: Enrichment (Editable) Intended Role: Instructor 11-3: Enrichment: Answer Key Intended Role: Instructor 11-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 11-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 11-4: Lesson Plan (PDF) Intended Role: Instructor 11-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 11-4 Intended Role: Instructor 11-4: Lesson Quiz (Editable) Intended Role: Instructor 11-4: Lesson Quiz: Answer Key Intended Role: Instructor 11-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 11-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 11-4: Additional Practice (Editable) Intended Role: Instructor 11-4: Additional Practice: Answer Key Intended Role: Instructor 11-4: Enrichment (Editable) Intended Role: Instructor 11-4: Enrichment: Answer Key Intended Role: Instructor 11-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 11-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 11: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 11: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 11: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 11: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 11: Assessment Form A (Editable) Intended Role: Instructor Topic 11: Assessment Form A: Answer Key Intended Role: Instructor Topic 11: Assessment Form B (Editable) Intended Role: Instructor Topic 11: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 12 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 12 Intended Role: Instructor Topic 12: Readiness Assessment (Editable) Intended Role: Instructor Topic 12: Readiness Assessment: Answer Key Intended Role: Instructor Topic 12: enVision STEM Masters: Answer Key Intended Role: Instructor 12-1: Lesson Plan (PDF) Intended Role: Instructor 12-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 12-1 Intended Role: Instructor 12-1: Lesson Quiz (Editable) Intended Role: Instructor 12-1: Lesson Quiz: Answer Key Intended Role: Instructor 12-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 12-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 12-1: Additional Practice (Editable) Intended Role: Instructor 12-1: Additional Practice: Answer Key Intended Role: Instructor 12-1: Enrichment (Editable) Intended Role: Instructor 12-1: Enrichment: Answer Key Intended Role: Instructor 12-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 12-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 12-2: Lesson Plan (PDF) Intended Role: Instructor 12-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 12-2 Intended Role: Instructor 12-2: Lesson Quiz (Editable) Intended Role: Instructor 12-2: Lesson Quiz: Answer Key Intended Role: Instructor 12-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 12-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 12-2: Additional Practice (Editable) Intended Role: Instructor 12-2: Additional Practice: Answer Key Intended Role: Instructor 12-2: Enrichment (Editable) Intended Role: Instructor 12-2: Enrichment: Answer Key Intended Role: Instructor 12-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 12-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 12-3: Lesson Plan (PDF) Intended Role: Instructor 12-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 12-3 Intended Role: Instructor 12-3: Lesson Quiz (Editable) Intended Role: Instructor 12-3: Lesson Quiz: Answer Key Intended Role: Instructor 12-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 12-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 12-3: Additional Practice (Editable) Intended Role: Instructor 12-3: Additional Practice: Answer Key Intended Role: Instructor 12-3: Enrichment (Editable) Intended Role: Instructor 12-3: Enrichment: Answer Key Intended Role: Instructor 12-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 12-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 12-4: Lesson Plan (PDF) Intended Role: Instructor 12-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 12-4 Intended Role: Instructor 12-4: Lesson Quiz (Editable) Intended Role: Instructor 12-4: Lesson Quiz: Answer Key Intended Role: Instructor 12-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 12-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 12-4: Additional Practice (Editable) Intended Role: Instructor 12-4: Additional Practice: Answer Key Intended Role: Instructor 12-4: Enrichment (Editable) Intended Role: Instructor 12-4: Enrichment: Answer Key Intended Role: Instructor 12-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 12-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 12-5: Lesson Plan (PDF) Intended Role: Instructor 12-5: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 12-5 Intended Role: Instructor 12-5: Lesson Quiz (Editable) Intended Role: Instructor 12-5: Lesson Quiz: Answer Key Intended Role: Instructor 12-5: Reteach to Build Understanding (Editable) Intended Role: Instructor 12-5: Reteach to Build Understanding: Answer Key Intended Role: Instructor 12-5: Additional Practice (Editable) Intended Role: Instructor 12-5: Additional Practice: Answer Key Intended Role: Instructor 12-5: Enrichment (Editable) Intended Role: Instructor 12-5: Enrichment: Answer Key Intended Role: Instructor 12-5: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 12-5: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 12: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 12: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 12: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 12: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 12: Assessment Form A (Editable) Intended Role: Instructor Topic 12: Assessment Form A: Answer Key Intended Role: Instructor Topic 12: Assessment Form B (Editable) Intended Role: Instructor Topic 12: Assessment Form B: Answer Key Intended Role: Instructor Teacher's Edition eText: Topic 13 Intended Role: Instructor Interactive Student Edition: Realize Reader: Topic 13 Intended Role: Instructor Topic 13: Readiness Assessment (Editable) Intended Role: Instructor Topic 13: Readiness Assessment: Answer Key Intended Role: Instructor Topic 13: enVision STEM Masters: Answer Key Intended Role: Instructor 13-1: Lesson Plan (PDF) Intended Role: Instructor 13-1: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 13-1 Intended Role: Instructor 13-1: Lesson Quiz (Editable) Intended Role: Instructor 13-1: Lesson Quiz: Answer Key Intended Role: Instructor 13-1: Reteach to Build Understanding (Editable) Intended Role: Instructor 13-1: Reteach to Build Understanding: Answer Key Intended Role: Instructor 13-1: Additional Practice (Editable) Intended Role: Instructor 13-1: Additional Practice: Answer Key Intended Role: Instructor 13-1: Enrichment (Editable) Intended Role: Instructor 13-1: Enrichment: Answer Key Intended Role: Instructor 13-1: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 13-1: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 13-2: Lesson Plan (PDF) Intended Role: Instructor 13-2: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 13-2 Intended Role: Instructor 13-2: Lesson Quiz (Editable) Intended Role: Instructor 13-2: Lesson Quiz: Answer Key Intended Role: Instructor 13-2: Reteach to Build Understanding (Editable) Intended Role: Instructor 13-2: Reteach to Build Understanding: Answer Key Intended Role: Instructor 13-2: Additional Practice (Editable) Intended Role: Instructor 13-2: Additional Practice: Answer Key Intended Role: Instructor 13-2: Enrichment (Editable) Intended Role: Instructor 13-2: Enrichment: Answer Key Intended Role: Instructor 13-2: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 13-2: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 13-3: Lesson Plan (PDF) Intended Role: Instructor 13-3: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 13-3 Intended Role: Instructor 13-3: Lesson Quiz (Editable) Intended Role: Instructor 13-3: Lesson Quiz: Answer Key Intended Role: Instructor 13-3: Reteach to Build Understanding (Editable) Intended Role: Instructor 13-3: Reteach to Build Understanding: Answer Key Intended Role: Instructor 13-3: Additional Practice (Editable) Intended Role: Instructor 13-3: Additional Practice: Answer Key Intended Role: Instructor 13-3: Enrichment (Editable) Intended Role: Instructor 13-3: Enrichment: Answer Key Intended Role: Instructor 13-3: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 13-3: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor 13-4: Lesson Plan (PDF) Intended Role: Instructor 13-4: Lesson Plan (Editable) Intended Role: Instructor Teacher's Edition eText: Lesson 13-4 Intended Role: Instructor 13-4: Lesson Quiz (Editable) Intended Role: Instructor 13-4: Lesson Quiz: Answer Key Intended Role: Instructor 13-4: Reteach to Build Understanding (Editable) Intended Role: Instructor 13-4: Reteach to Build Understanding: Answer Key Intended Role: Instructor 13-4: Additional Practice (Editable) Intended Role: Instructor 13-4: Additional Practice: Answer Key Intended Role: Instructor 13-4: Enrichment (Editable) Intended Role: Instructor 13-4: Enrichment: Answer Key Intended Role: Instructor 13-4: Mathematical Literacy and Vocabulary (Editable) Intended Role: Instructor 13-4: Mathematical Literacy and Vocabulary: Answer Key Intended Role: Instructor Topic 13: Performance Assessment Form A (Editable) Intended Role: Instructor Topic 13: Performance Assessment Form A: Answer Key Intended Role: Instructor Topic 13: Performance Assessment Form B (Editable) Intended Role: Instructor Topic 13: Performance Assessment Form B: Answer Key Intended Role: Instructor Topic 13: Assessment Form A (Editable) Intended Role: Instructor Topic 13: Assessment Form A: Answer Key Intended Role: Instructor Topic 13: Assessment Form B (Editable) Intended Role: Instructor Topic 13: Assessment Form B: Answer Key Intended Role: Instructor End-of-Course Assessment (Editable) Intended Role: Instructor End-of-Course Assessment: Answer Key Intended Role: Instructor Mathematics II Next-Generation Practice Test: Answer Key Intended Role: Instructor 3/4-Year Performance Task 1: Answer Key Intended Role: Instructor 3/4-Year Performance Task 2: Answer Key Intended Role: Instructor Teacher's Edition eText: Mathematics II Intended Role: Instructor eText Container Interactive Student Edition: Realize Reader: Mathematics II Tools Glossary Glossary Desmos Graphing Calculator Math Tools Glossary Desmos Scientific Calculator Desmos Graphing Calculator Desmos Geometry Glossary Math Tools Math Tools Glossary Desmos Graphing Calculator Desmos Graphing Calculator Desmos Geometry Desmos Geometry Desmos Geometry Desmos Graphing Calculator Desmos Geometry Desmos Graphing Calculator Math Tools Desmos Scientific Calculator Desmos Geometry Math Tools Desmos Scientific Calculator Desmos Geometry Desmos Graphing Calculator Math Tools Desmos Graphing Calculator Desmos Graphing Calculator Glossary Math Tools Desmos Geometry Desmos Graphing Calculator Desmos Geometry Math Tools Glossary Glossary Math Tools Desmos Scientific Calculator Desmos Scientific Calculator Desmos Scientific Calculator Desmos Scientific Calculator Glossary Desmos Scientific Calculator Desmos Geometry Desmos Geometry Glossary Math Tools Math Tools Desmos Scientific Calculator Glossary Desmos Graphing Calculator Desmos Scientific Calculator Desmos Scientific Calculator