Organization: McGraw-Hill Product Name: Reveal Geometry 2020 Product Version: 1.2 Source: IMS Online Validator Profile: 1.2.0 Identifier: cc_v1p2 Timestamp: Monday, September 14, 2020 09:25 AM EDT Status: VALID! Conformant: true ----- VALID! ----- Resource Validation Results The document is valid. ----- 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: MAFS.6.NS.3.6.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. - 4f90be5f-2d92-4c88-bc18-34d3874d4075 MAFS.K12.MP.6 Attend to precision. - 340833b6-d1c4-44eb-9647-f79f7daf0030 MAFS.912.F-IF.2.6 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. - 1d1f44f9-3b27-4ba9-8a74-8cf374455d45 MAFS.7.G.2 Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. - 61262f84-974b-4139-bf59-3b054be2f569 CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers. - 1E506492-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - 4593061d-54f3-40e8-a8ea-361468c7e129 MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite it. - ab53e453-944a-43b3-9178-8443bbd95b16 MAFS.912.A-REI.2.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - 74b31224-ff95-4c0d-a7d3-3ed7d443ee48 MAFS.912.S-MD.2.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). - af3d22fc-b1a1-409a-87c2-8b17e6e8e1aa CCSS.Math.Content.HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. - 20E5E0A6-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-C.B.5 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. - 20EBBDD2-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.NS.1 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. - 9aadcaf3-9c87-43e4-85ac-94bba7329393 CCSS.Math.Content.4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. - 1D8D6FD2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions. - 1E5EFE76-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.SP.C.8.a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 1F11763C-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.G.A Solve real-world and mathematical problems involving area, surface area, and volume. - 1E841F08-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. - 1EDE4C8A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar. - 20E4309E-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.SP.1 Investigate patterns of association in bivariate data. - 6fc197ec-34df-46f6-8b8b-08338bfeb315 CCSS.Math.Content.HSF-IF.B.6 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. - 202DFCA2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.A.1.c Parallel lines are taken to parallel lines. - 1F57380C-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. - 1ECF626A-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.2.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - af3ceb2b-9f8c-40fc-a591-465675ef4e60 CCSS.Math.Content.7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. - 1EDCE3A4-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-CO.B.7 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. - 20BA8D52-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - 210DFEC4-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-C.A.4 Construct a tangent line from a point outside a given circle to the circle. - 20E8E774-7053-11DF-8EBF-BE719DFF4B22 MAFS.6.RP.1.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. - 94c59a18-9a16-401a-9457-d6a22f3be8a0 CCSS.Math.Content.8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - 1F59A7EA-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - cba144bc-3a8b-400f-a8b9-68bee0dd9926 MAFS.8.G.1.1.b Angles are taken to angles of the same measure. - 903ce2cd-7e8c-4356-bf7b-367e86bacae1 MAFS.912.G-SRT.4.10 Prove the Laws of Sines and Cosines and use them to solve problems. - 56c0ac6d-5fe0-4719-b8e4-64c579119fab CCSS.Math.Content.HSG-GMD.A.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. - 210355DC-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.SP.C.7.a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. - 1F094F8E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: - 1F530372-7053-11DF-8EBF-BE719DFF4B22 MAFS.6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. - 50458c9e-ed0f-4cba-a28a-f4957022f614 MAFS.912.G-CO.1.5 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. - f3f3fa71-e1fe-457d-a89a-df6cb328c1aa MAFS.912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - 9230244b-1de4-45d0-ae46-d330edf0fe97 CCSS.Math.Content.6.RP.A Understand ratio concepts and use ratio reasoning to solve problems. - 1E1E82CE-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.F.2.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (𝘹, 𝘹𝘺) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 7ecdef3a-4724-47df-9619-c69740336ee3 CCSS.Math.Content.HSG-CO.C.10 Prove theorems about triangles. - 20C0B0A6-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - 20B467E2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GPE Expressing Geometric Properties with Equations - 20ED034A-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.2 Reason abstractly and quantitatively. - 3885dc10-69f7-4a9d-b273-ed83a569094a MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. - 4520de54-8e98-4005-89b2-785e033115e9 CCSS.Math.Content.HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - 20D53E36-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.G.1.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - d7653434-a0cf-4e26-a1ba-0409fd00c56e MAFS.7.SP.3.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. - 085ee8bd-4a44-46ee-97af-5253b05b064c CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning. - A6497ED0-6F89-11DF-BAEE-EA329DFF4B22 MAFS.6.NS.3.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. - e3d729ff-7887-499d-bba7-0eb4a3bfd06a CCSS.Math.Content.8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 1F624D6E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. - 1F6E148C-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. - 1FC900FE-7053-11DF-8EBF-BE719DFF4B22 Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝘹𝘺𝑥² + 9𝘹𝘺𝑥𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝘹𝘺𝑥𝑥𝑥 – 𝘹𝘺𝑥𝑥𝑥𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝘹𝘺𝑥𝑥𝑥𝑦𝑥 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦. - 5C5688E4-7377-11DF-A1E8-223D9DFF4B22 MAFS.7.G.2.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 0285a2c7-5878-4e53-ba5a-7fa43a304ba3 MAFS.912.G-C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. - 496eb64d-a4b3-4eab-87cc-1468af8f8f00 CCSS.Math.Content.7.SP.C Investigate chance processes and develop, use, and evaluate probability models. - 1F018E7A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. - 1F0453C6-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.3.1 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. - 78332818-27d5-4b06-9369-fd35deb0af11 CCSS.Math.Content.7.NS.A.1.c Understand subtraction of rational numbers as adding the additive inverse, 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱 – 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲 = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱 + (–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 1ECD20F4-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - 20FD160E-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.G.1 Draw, construct, and describe geometrical figures and describe the relationships between them. - 77ac194b-7098-4bc9-b094-873db3a95c08 Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. - 5C561224-7377-11DF-A1E8-223D9DFF4B22 CCSS.Math.Content.HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - 20FE19D2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. - 1EC7A304-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. - 21793be4-f5f1-4d68-a8d1-fe1fa0bcdfd5 CCSS.Math.Content.HSG-CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). - 20B08780-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - 20C6B74E-7053-11DF-8EBF-BE719DFF4B22 Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. - 5C551ED2-7377-11DF-A1E8-223D9DFF4B22 CCSS.Math.Content.HSG-CO.C.11 Prove theorems about parallelograms. - 20C1DF58-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. - 1E28AC4A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSS-CP.B.6 Find the conditional probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈 given 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉 as the fraction of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉’s outcomes that also belong to 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - 2147AE26-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. - 1ED58500-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.NS.C.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. - 1E517FEE-7053-11DF-8EBF-BE719DFF4B22 MAFS.6.G.1 Solve real-world and mathematical problems involving area, surface area, and volume. - 449e018b-2328-413f-b91e-dbd0864c9696 MAFS.8.EE.1 Work with radicals and integer exponents. - 56c44c41-66d7-4862-aa7a-4f89eb3d2fe2 CCSS.Math.Content.HSG-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems. - 20DE5520-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.2.6 Explain a proof of the Pythagorean Theorem and its converse. - 4eb06f5a-f0dd-4acd-97b4-cfd5c4f536e3 MAFS.8.G.3 Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. - 941cba33-148d-46ed-9d6d-5293f5522941 MAFS.6.RP.1.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. - 8d888642-08c1-45e3-8286-0420933ab148 MAFS.912.G-MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). - 5cac886f-fb66-4375-955c-ca4530b395fe MAFS.7.SP.3.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 8c2e34cc-06da-4f47-b0d7-52085dae3c43 CCSS.Math.Content.HSS-CP.A.3 Understand the conditional probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈 given 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉 as 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉)/𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - 213E236A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.F-LE.2.5 Interpret the parameters in a linear or exponential function in terms of a context. - 3d9245a7-1c9a-4797-80f5-9265839bcc7d CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹, 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 1F4E8D1A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. - 234e8170-328a-447a-a55e-73773059ea52 MAFS.912.G-SRT.1.2 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. - a3c4ea87-e07a-4026-adfe-63b90cfa1990 CCSS.Math.Content.6.RP.A.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. - 1E331C2A-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.SP.1.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. - 2ed72bc0-754f-4f10-be2e-0113c528a261 MAFS.K12.MP.7.1 Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥² + 9𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥 – 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦. - 753f388e-e7b0-4192-b79d-4dac6f0ca194 MAFS.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - a02afe23-a8a7-4648-b698-f2552414f99e MAFS.7.NS.1.2.b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲 are integers, then –(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱/𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲) = (–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱)/𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲 = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱/(–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲). Interpret quotients of rational numbers by describing real-world contexts. - 5526b516-9ec3-4ede-8383-1bb5811b7875 CCSS.Math.Content.HSG-SRT.D.11 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). - 20DFAD30-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. - 1EC61534-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.8.1 Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦 – 2)/(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥 – 1)(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥 + 1), (𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥 – 1)(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. - 464690b2-2313-4a93-a9da-ea512f6b7ed6 CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. - 1E20776E-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-GPE.1.1 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. - e139b145-463b-427d-af84-a5ac07fce347 MAFS.K12.MP.5 Use appropriate tools strategically. - 7fb94672-da71-44b5-950f-b57258b31b63 CCSS.Math.Content.HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. - 1F843D84-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.SP.3.8.a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 8c847e27-fc22-4af3-8b50-96212dee2969 CCSS.Math.Content.6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. - 1E86309A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-C.1.4 Construct a tangent line from a point outside a given circle to the circle. - 70507dc3-489f-43b1-af92-2e51580b15f6 MAFS.912.S-CP.1.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. - 07e659dc-92c0-41d5-b63e-da5d04dfaf66 CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. - A6430F50-6F89-11DF-BAEE-EA329DFF4B22 CCSS.Math.Content.7.NS.A.2.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - 1ED49A78-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.1.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - 04b1cf07-e06f-4356-97f8-7938f054152c MAFS.7.NS.1.1.b Understand 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱 + 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲 as the number located a distance |𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲| from 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱, in the positive or negative direction depending on whether 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲 is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - d2719f8b-2283-4c45-9ef0-0a8687806d67 MAFS.6.RP.1.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. - c49ed342-fc0a-4f9d-a8f0-8428508960be CCSS.Math.Content.8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 1F665990-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.6.1 Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. - cc88d0e7-1375-41f3-89ff-b0557b54b359 MAFS.8.G.2 Understand and apply the Pythagorean Theorem. - 5563d779-eefa-4070-add9-e14790b67f0f CCSS.Math.Content.6.NS.C.7.c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. - 1E5725AC-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. - e8ccb2a6-4a82-43e7-913d-bb47974127d5 MAFS.7.NS.1.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. - 12f9d784-82f8-48df-bf18-e91c217cb151 CCSS.Math.Content.HSG-CO.A.5 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. - 20B61254-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 1EF1A00A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSS-CP.B.7 Apply the Addition Rule, 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈 or 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉) = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈) + 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - 214BA0DA-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.1.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). - 365e2305-0416-4ba3-8680-d56ec4c87bda MAFS.912.S-CP.2.8 Apply the general Multiplication Rule in a uniform probability model, 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉) = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈)𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉|𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈) = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉)𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈|𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - 6226df4c-7bd4-42a9-878d-be96a1ac2ebd MAFS.6.NS.3.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. - 26d5bb75-cfc7-4425-bc12-10308d10e484 CCSS.Math.Content.HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). - 213B2728-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. - A630F7A2-6F89-11DF-BAEE-EA329DFF4B22 CCSS.Math.Content.7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 1ED93CAE-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. - 1FF53976-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A.2.b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱 and 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲 are integers, then –(𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱/𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲) = (–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱)/𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲 = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱/(–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲). Interpret quotients of rational numbers by describing real-world contexts. - 1ED1FDF4-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Practice.MP7 Look for and make use of structure. - A648672A-6F89-11DF-BAEE-EA329DFF4B22 MAFS.912.G-GPE.1.2 Derive the equation of a parabola given a focus and directrix. - 3c35b595-e855-489b-9ea9-5217bf140b81 CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 1F611A98-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.A.1.b Angles are taken to angles of the same measure. - 1F55D2E6-7053-11DF-8EBF-BE719DFF4B22 MAFS.6.NS.3.6.b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - b30cc0e2-e28c-4024-aa51-4090f5b9beea MAFS.912.G-GMD.2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - b94dc896-fb31-494c-8dbb-56622445c6dc MAFS.8.G.1.1.a Lines are taken to lines, and line segments to line segments of the same length. - f57cdb34-591b-4951-bb62-41be911fd396 CCSS.Math.Content.HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). - 210F2E7A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.SP.C.7.b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. - 1F0C8E56-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.2.1 Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. - 0331111e-4349-473a-ae23-5b4b6572bb0c MAFS.7.SP.3.8.b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. - 46d4ded1-2d2e-475d-ae22-4655823b5aac CCSS.Math.Content.HSG-GPE.A.1 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. - 20F02CFA-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). - 95771a27-cf28-4500-8649-771dcbacf285 Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. - 5C559BA0-7377-11DF-A1E8-223D9DFF4B22 MAFS.912.G-GPE.2.5 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). - 84287e32-275f-426c-a455-a831b54f0cbe MAFS.7.NS.1.1.d Apply properties of operations as strategies to add and subtract rational numbers. - dd623b17-83ce-4a1f-8bd3-32ebc397f31d MAFS.7.NS.1.3 Solve real-world and mathematical problems involving the four operations with rational numbers. - c540025f-5167-4f59-b73a-67d8eb49011a MAFS.8.G.1.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. - 2976f57c-63d7-4eb5-9048-26ff05c61bfe CCSS.Math.Content.HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - 20D0CCA2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. - 206AD910-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers. - 1E4523FC-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.NS.1.1.c Understand subtraction of rational numbers as adding the additive inverse, 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱 – 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲 = 𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱 + (–𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘈𝘉𝘉𝘈𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘹𝘺𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 29db58cc-f23f-4cdb-bb9b-ad7acaf9f8f2 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. - 5C54A65A-7377-11DF-A1E8-223D9DFF4B22 MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others. - 0ab2c4e3-2b87-4dd9-a12b-353a486762fa CCSS.Math.Content.7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. - 1EEF675E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. - 1F26B254-7053-11DF-8EBF-BE719DFF4B22 Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. - 5C53AC00-7377-11DF-A1E8-223D9DFF4B22 CCSS.Math.Content.HSG-SRT.B.4 Prove theorems about triangles. - 20D3E716-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. - 1E4879DA-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Practice.MP6 Attend to precision. - A647016E-6F89-11DF-BAEE-EA329DFF4B22 CCSS.Math.Content.5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties. - 1E0D090E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.NS.C.6.b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - 1E4CFA28-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GMD.A.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - 21058848-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 1EF04980-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). - e2251658-c086-4a0b-a646-c38b02571bfb CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. - 1EE9F0EE-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.1.3 Understand the conditional probability of 34𝘈 given 34𝘈𝘉 as 34𝘈𝘉𝘗(34𝘈𝘉𝘗𝘈 and 34𝘈𝘉𝘗𝘈𝘉)/34𝘈𝘉𝘗𝘈𝘉𝘗(34𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - 15e241f3-e466-4c02-9ccd-1e2c64a44542 CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). - 211096B6-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.1.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. - d1e351aa-2fe7-42a5-86cd-50ac103b8085 MAFS.912.G-MG.1.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). - d34a51ae-323a-4fbe-8f91-eb4df598d96a MAFS.912.G-SRT.3.6 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. - 313f2846-7fe7-4707-b21e-d95e6904380f CCSS.Math.Content.HSG-CO.A.1 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. - 20AF0F68-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.EE.2.4.a Solve word problems leading to equations of the form 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹 + 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲 = 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳 and 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱(34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹 + 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲) = 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 535bbd08-5ee8-43b0-9eaa-036060b2a38d MAFS.7.SP.3.7.a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. - 644c73bb-c2a7-4598-a13b-5630c22ef333 CCSS.Math.Content.8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software. - 1F527D08-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GPE.B.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). - 20FB82F8-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - 8081346d-4e11-4855-bb92-5327b015f168 MAFS.8.G.1 Understand congruence and similarity using physical models, transparencies, or geometry software. - d20d06e9-6442-49a0-8f16-bd94d3cc66b0 CCSS.Math.Content.HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - 200598A2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. - 1D94061C-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - 20E77AEC-7053-11DF-8EBF-BE719DFF4B22 MAFS.6.NS.3 Apply and extend previous understandings of numbers to the system of rational numbers. - ce84cbd5-eac0-4b0b-bf6c-4311036c7d9f CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse. - 1F5F88B8-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.EE.B.4.a Solve word problems leading to equations of the form 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹 + 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲 = 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳 and 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱(34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹 + 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲) = 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 34𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 1EE190CA-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.1.1 Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. - db656dba-3ee7-423e-9020-626812d8c8d3 CCSS.Math.Content.HSG-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - 20BC6622-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.EE.A Work with radicals and integer exponents. - 1F2530C8-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix. - 20F1B6C4-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. - f133b6ed-fc63-41e5-a15e-aa0517226554 MAFS.912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. - 6a70edab-fdb9-4fe6-a5e9-65c49f2f2f7d CCSS.Math.Content.8.G.A.1.a Lines are taken to lines, and line segments to line segments of the same length. - 1F5487CE-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - 1F5A8638-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.NS.A.1.b Understand 34𝘱 + 34𝘱𝘲 as the number located a distance |34𝘱𝘲𝘲| from 34𝘱𝘲𝘲𝘱, in the positive or negative direction depending on whether 34𝘱𝘲𝘲𝘱𝘲 is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - 1ECBB426-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.1.1.c Parallel lines are taken to parallel lines. - 5b59eea4-2f9e-4e0c-be6f-6387157c2fc6 CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 1EEB2176-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - 01592a48-b9ca-4802-b5f1-459e8b71a6ca CCSS.Math.Content.HSG-SRT.C.6 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. - 20D81638-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. - 1F642008-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.G.1.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - f5938db7-c588-435c-b840-c712b7673dab CCSS.Math.Content.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - 1EE09E22-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSS-CP.A.2 Understand that two events 34𝘱𝘲𝘲𝘱𝘲𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉 are independent if the probability of 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - 213C6F7A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). - 20C48CD0-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.2.9 Use permutations and combinations to compute probabilities of compound events and solve problems. - c0c384a2-4fc8-426a-8c3b-03dbf808fbb8 MAFS.8.G.2.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 4e00cff0-0a3f-4199-8b2a-11056c4c5b6a MAFS.6.NS.3.7 Understand ordering and absolute value of rational numbers. - a928c4a0-7ce2-40ac-b529-b03919c04efa MAFS.7.NS.1.2.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - 02e76175-0331-4e07-aa87-eda20ed69cde MAFS.8.G.3.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 6b3e99c5-9d62-4b00-a8a2-909ca3c21f17 CCSS.Math.Content.7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - 1EECC468-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - 0c7baec3-56e0-4d9c-a553-58a39e0184c5 Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. - 5C5422D4-7377-11DF-A1E8-223D9DFF4B22 CCSS.Math.Content.HSG-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. - 20F8538A-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.G.B Understand and apply the Pythagorean Theorem. - 1F5E1C9E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺 = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹 for a line through the origin and the equation 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺 = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹 + 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣 for a line intercepting the vertical axis at 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣. - 1F34E2FC-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.2.6 Find the conditional probability of 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈 given 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉 as the fraction of 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉’s outcomes that also belong to 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - 1d933095-b158-4d60-923c-5472150d0da2 CCSS.Math.Content.HSG-SRT.A.2 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. - 20CF6E2A-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.EE.2.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. - 9363a0ab-0c0e-42c8-8087-c1d47e142616 MAFS.K12.MP.7 Look for and make use of structure. - 4b6d6fc7-f9ff-4e3c-89c9-3642b61650a0 MAFS.8.SP.1.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. - 5c7985c2-455b-4609-a53f-3c6a4ce8ca7a CCSS.Math.Content.6.NS.C.6.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. - 1E4EBC5A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.A-CED.1.3 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. - 3940bed0-00a3-49dc-9b74-172f6db23b03 MAFS.8.G.1.1 Verify experimentally the properties of rotations, reflections, and translations: - beb495c9-71fc-4e42-bb92-718ed1466832 Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦 – 2)/(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥 + 1), (34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥² + 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. - 5C56FF86-7377-11DF-A1E8-223D9DFF4B22 MAFS.7.SP.3 Investigate chance processes and develop, use, and evaluate probability models. - 42650adb-825c-478c-8398-ffc653c7f412 CCSS.Math.Content.8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. - 1F715E62-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: - 1D873748-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-SRT.1.1 Verify experimentally the properties of dilations given by a center and a scale factor: - 21569d11-ff4f-458a-8c88-8d5ae94eb354 MAFS.K12.MP.1 Make sense of problems and persevere in solving them. - efdcc5ef-fe77-460b-b67c-303a7528d39d CCSS.Math.Content.HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. - 20D9571E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - 21097BCE-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-CO.C.9 Prove theorems about lines and angles. - 20BF272C-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. - 1E629B3A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - 5e7c7b2c-bced-4180-82b4-31c50afc3434 MAFS.7.G.2.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 81aadb74-cffc-4158-a0c7-76113c625ede CCSS.Math.Content.HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - 20B27086-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSG-SRT.D.9 Derive the formula 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴 = 1/2 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣 sin(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. - 20DD6142-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.8.SP.A Investigate patterns of association in bivariate data. - 1F69AE60-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.EE.2.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - d4254c8a-34a7-43af-90c9-cf2612cd0b0f MAFS.7.EE.2 Solve real-life and mathematical problems using numerical and algebraic expressions and equations. - 84704cc3-a1f2-40ad-8699-ea628597af61 MAFS.6.EE.1.2 Write, read, and evaluate expressions in which letters stand for numbers. - d9778b4b-9a14-4e70-ad22-d0e40da9bc0c MAFS.912.G-C.1.1 Prove that all circles are similar. - 3f5fa520-fb4e-4aac-924a-30c9101b37d8 MAFS.8.EE.1.2 Use square root and cube root symbols to represent solutions to equations of the form 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹² = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹³ = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱, where 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 31922f5c-2d13-4455-9775-6785a6e9340a CCSS.Math.Content.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - 20098A98-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.A-REI.1.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - cb1cbbc6-8b32-4339-bc18-7439326ebb04 CCSS.Math.Content.HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - 2107150A-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.S-CP.2.7 Apply the Addition Rule, 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈 or 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉) = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈) + 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - 6427d00f-ad6e-4455-9421-b19c8a2232d0 CCSS.Math.Content.7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. - 1F0FB072-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.8 Look for and express regularity in repeated reasoning. - 6ff0dba2-4ea2-454f-8c62-e326c5d5841a CCSS.Math.Content.6.EE.A.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). - 1E6B15B2-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.1.1 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. - 3dccb19b-3ace-49cc-9857-0333e027e9ce MAFS.912.N-Q.1.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. - cc0687ff-bd03-4206-bfd1-a6c00a3b0e6d CCSS.Math.Content.8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. - 1F5BA5FE-7053-11DF-8EBF-BE719DFF4B22 MAFS.K12.MP.4 Model with mathematics. - f68eaa41-9ef2-4f61-941b-e67b099f9c02 CCSS.Math.Content.HSS-CP.B.8 Apply the general Multiplication Rule in a uniform probability model, 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉) = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈)34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉|34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈) = 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉)34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗(34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈|34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - 21512514-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSS-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). - 21C4A7FA-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. - 1E8B99B8-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. - 1EC1E018-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.2.7 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. - 6c776615-b919-4398-92f9-6099fa3e8534 CCSS.Math.Practice.MP4 Model with mathematics. - A6446404-6F89-11DF-BAEE-EA329DFF4B22 MAFS.K12.MP.4.1 Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. - e779da50-d12e-4314-966c-08e1656d94a3 CCSS.Math.Content.HSS-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems. - 2152A6E6-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 1F07DFD2-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. - 1EEE21BE-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - fbfc9efe-da5a-4183-befd-a59c09d46a2a MAFS.912.G-GMD.1.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - e9c4e09c-e06c-4624-956c-09440b6136d9 MAFS.912.S-MD.2.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). - b551a069-e77a-4830-97c4-f82209c14f6b MAFS.912.S-CP.1.2 Understand that two events 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 are independent if the probability of 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘈 and 34𝘱𝘲𝘲𝘱𝘲𝘈𝘉𝘈𝘉𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝐴𝘢𝘣𝐶𝘹𝘱𝘹𝘱𝘱𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - 7fda3047-95b7-4b43-b537-70f69e6e0338 CCSS.Math.Content.7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. - 1EE8951E-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.6.SP.B.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. - 1EA15910-7053-11DF-8EBF-BE719DFF4B22 MAFS.912.G-CO.2.6 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. - 12cad093-3392-4c22-92b8-94786c64a135 MAFS.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. - 0895ac3a-a4af-48a4-adee-700e9bf4b6b7 CCSS.Math.Content.8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - 1F585250-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.HSS-CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. - 213F8494-7053-11DF-8EBF-BE719DFF4B22 CCSS.Math.Content.4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. - 1D915D72-7053-11DF-8EBF-BE719DFF4B22 MAFS.7.SP.3.7.b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. - 0ac5d829-fc1d-4d6d-84a1-ed2ba7e383f6 MAFS.912.G-GPE Geometry: Expressing Geometric Properties with Equations - b39e9429-acd2-4582-adb0-5de334dc61c0 MAFS.7.NS.1.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. - 9daa3dc6-d802-454b-a275-54aa12fd5b15 MAFS.6.G.1.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. - 416ec9a2-b7d9-4c7f-a7d7-8701f5e201c3 MAFS.6.EE.1.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). - 5cefdb32-bf7b-4ff2-992d-f9aa633d8b36 MAFS.912.G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - 059590f5-e505-475d-b4a1-0fa470189fc2 MAFS.7.G.2.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. - 738d983c-11f3-44a9-805a-52d25669dbd6 MAFS.7.SP.3.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. - e8a44e9d-0501-4545-ace7-fefae7bb7c69 CCSS.Math.Content.HSS-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). - 21C63354-7053-11DF-8EBF-BE719DFF4B22 MAFS.8.G.1.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - ec01f684-e782-4cac-a4be-413821373d3