Organization: McGraw-Hill
Product Name: Reveal Math Course 2 2020
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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.6.SP.2.5.c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. - 6a475527-2c2f-42e6-a0c8-cd8bccfb8bcc
CCSS.Math.Content.6.EE.B Reason about and solve one-variable equations and inequalities. - 1E746F4A-7053-11DF-8EBF-BE719DFF4B22
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
6.NS.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. - C548DDAA-96FF-11E0-9509-C03D9DFF4B22
MAFS.7.NS.1.2.c Apply properties of operations as strategies to multiply and divide rational numbers. - bf13f75d-893d-4a97-85ac-2145bc938b46
7.DSP.5 Investigate the concept of probability of chance events. - 0908acdb-7852-4bbe-8abb-d20343271971
CCSS.Math.Content.7.NS.A.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - 1ED0C7D6-7053-11DF-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.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.7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. - 1EFE4E72-7053-11DF-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.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. - 1EFA6C26-7053-11DF-8EBF-BE719DFF4B22
MAFS.7.RP.1.2.d Explain what a point (𝘹, 𝘹𝘺) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, 𝘹𝘺𝘳) where 𝘹𝘺𝘳𝘳 is the unit rate. - d1f5b0d2-bbd8-4659-a97e-54a97b6d78dc
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
CCSS.Math.Content.7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. - 1EDCE3A4-7053-11DF-8EBF-BE719DFF4B22
MAFS.7.NS.1.1.a Describe situations in which opposite quantities combine to make 0. - 19209fd0-ecde-46cc-bef1-8054d559d3aa
8.A model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas; - 9F0A0C68-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.7.EE.1 Use properties of operations to generate equivalent expressions. - 0527d2ad-3dc8-46e5-b100-9b8e3c8f745f
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
MAFS.6.EE.2.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. - 6b6dcc4d-86c4-402d-8de8-34d562c78491
PFA.6.13 The student will solve one-step linear equations in one variable, including practical problems that require the solution of a one-step linear equation in one variable. - BFA63968-C66C-11E6-9282-BAECCCC8CA83
CCSS.Math.Content.6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. - 1E95F282-7053-11DF-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.6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: - 1E9CD886-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.6.SP.B.5.d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. - 1EA69704-7053-11DF-8EBF-BE719DFF4B22
7.RP.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - C56755B4-96FF-11E0-9509-C03D9DFF4B22
MAFS.6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. - 50458c9e-ed0f-4cba-a28a-f4957022f614
MAFS.6.G.1.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas 𝘹𝘺𝘳𝘳𝘝 = 𝘹𝘺𝘳𝘳𝘝𝘭 𝘹𝘺𝘳𝘳𝘝𝘭𝘸 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩 to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - 4a6e5924-f5d3-4474-a770-cec97cbdbee5
CCSS.Math.Content.6.RP.A Understand ratio concepts and use ratio reasoning to solve problems. - 1E1E82CE-7053-11DF-8EBF-BE719DFF4B22
7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. - C5815586-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.7.EE.A Use properties of operations to generate equivalent expressions. - 1ED7DB02-7053-11DF-8EBF-BE719DFF4B22
MG.7.4.a describe and determine the volume and surface area of rectangular prisms and cylinders; and - 48FDDB7E-C723-11E6-806E-51E7CCC8CA83
CCSS.Math.Content.7.RP.A.2.c Represent proportional relationships by equations. - 1EBD422E-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.6.NS.B.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. - 1E419336-7053-11DF-8EBF-BE719DFF4B22
MAFS.K12.MP.2 Reason abstractly and quantitatively. - 3885dc10-69f7-4a9d-b273-ed83a569094a
CCSS.Math.Content.6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents. - 1E5FE70A-7053-11DF-8EBF-BE719DFF4B22
7.DSP.6.c Compare theoretical and experimental probabilities. - ad718c81-f64a-4dde-aaeb-5a3addea5435
MAFS.6.NS.2 Compute fluently with multi-digit numbers and find common factors and multiples. - c1801500-4a96-4f5a-88fc-259110d5d691
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
11.A model and solve one-variable, two-step equations and inequalities; - 9F1051AE-0D0A-11E2-9583-8B2E9DFF4B22
3.D add, subtract, multiply, and divide integers fluently; and - 9EE02D1C-0D0A-11E2-9583-8B2E9DFF4B22
11.C write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. - 9F115D74-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.SP.1.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. - 7a07f2a6-b564-40d2-aa66-e48c4fbd09af
7.NS.3 Apply the concepts of all four operations with rational numbers to solve real-world and mathematical problems. - be0b5751-6dff-4c27-8da7-654d84453e79
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
7.RP.2.a Determine when two quantities are in a proportional relationship. - c24c0201-4842-4445-88c2-57dd487bc8ac
CCSS.Math.Content.7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems. - 1EB29FEA-7053-11DF-8EBF-BE719DFF4B22
7.GM.6 Apply the concepts of two- and three-dimensional figures to real-world and mathematical situations. - 55869c2c-3c2b-42a6-ad14-121eca11e424
7.DSP.5.d Understand that a probability closer to 1 indicates a likely chance event. - c5fcaa2a-b202-4f3a-841a-86af941b0cce
CCSS.Math.Content.7.RP.A.2.d Explain what a point (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳) where 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳 is the unit rate. - 1EC08E02-7053-11DF-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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥² + 9𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥 + 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(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥 – 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦)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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦. - 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
CCSS.Math.Content.6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩 to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - 1E87888C-7053-11DF-8EBF-BE719DFF4B22
10.B represent solutions for one-variable, two-step equations and inequalities on number lines; and - 9F0ED75C-0D0A-11E2-9583-8B2E9DFF4B22
CCSS.Math.Content.7.SP.C Investigate chance processes and develop, use, and evaluate probability models. - 1F018E7A-7053-11DF-8EBF-BE719DFF4B22
MAFS.6.EE.1.1 Write and evaluate numerical expressions involving whole-number exponents. - 43c4ba36-948f-4d06-8ff6-374622a2814d
MG.8.8 The student will construct a three-dimensional model, given the top or bottom, side, and front views. - 95343B80-C725-11E6-9AF9-0D71BF03DF2F
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
MAFS.7.SP.3.8.c Design and use a simulation to generate frequencies for compound events. - ca3d36da-490f-49db-bb51-b0905e575762
CCSS.Math.Content.7.NS.A.1.c Understand subtraction of rational numbers as adding the additive inverse, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱 – 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱 + (–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲). 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
PFA.6.12.a represent a proportional relationship between two quantities, including those arising from practical situations; - 9B4B9D42-C66C-11E6-91EA-BE72BF03DF2F
MAFS.6.RP.1.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. - b9e527ce-d1c9-4684-8ba7-00cedec7c152
MAFS.7.G.1 Draw, construct, and describe geometrical figures and describe the relationships between them. - 77ac194b-7098-4bc9-b094-873db3a95c08
CCSS.Math.Content.7.RP.A.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 1EBB74C6-7053-11DF-8EBF-BE719DFF4B22
7.GM.4.d Use the formulas for circumference and area of circles appropriately to solve real-world and mathematical problems. - e17fed29-681d-434c-950f-d2c40f25d1f9
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
MAFS.7.SP.2.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. - a5b63582-a920-4e7c-9f2d-353a594a5274
8.A extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle; - 9EEC3F08-0D0A-11E2-9583-8B2E9DFF4B22
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.7.EE.2.4.b Solve word problems leading to inequalities of the form 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲 > 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳 or 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲 < 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. - 40fa43b7-fd9f-4d42-851d-04243eb7f8ca
7.NS.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. - C56B37A6-96FF-11E0-9509-C03D9DFF4B22
11.B determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points; and - 9F300800-0D0A-11E2-9583-8B2E9DFF4B22
7.DSP.6 Investigate the relationship between theoretical and experimental probabilities for simple events. - 2a47e83e-780d-4c55-ae9f-8e7ff485af77
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.6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions. - 1E371E1A-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. - 1E3FE838-7053-11DF-8EBF-BE719DFF4B22
7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. - C5834A44-96FF-11E0-9509-C03D9DFF4B22
PFA.7.11 The student will evaluate algebraic expressions for given replacement values of the variables. - 67B6A20C-C724-11E6-AB09-BEE8CCC8CA83
7.GM.6.a Understand that the concept of area is applied to two-dimensional figures such as triangles, quadrilaterals, and polygons. - 9a8e2df3-f403-4dd5-91a8-e16c3297d870
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.7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. - 1ED58500-7053-11DF-8EBF-BE719DFF4B22
4.B represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity; - 9ECAB9AA-0D0A-11E2-9583-8B2E9DFF4B22
7.DSP.7.b Develop both uniform and non-uniform probability models. - e5180b3a-fc16-4559-b4b7-0253508c07d4
MAFS.7.EE.1.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. - 99349299-82c2-4425-95b3-ecdcf299d99b
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
7.DSP.5.b Understand that probability measures likelihood of a chance event occurring. - d77f7c87-650c-406e-8f9c-071c23c920e6
CCSS.Math.Content.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢 ÷ 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. - 1DD01CA6-7053-11DF-8EBF-BE719DFF4B22
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
CCSS.Math.Content.7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. - 1F02DA28-7053-11DF-8EBF-BE719DFF4B22
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.6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. - 1E9B5C72-7053-11DF-8EBF-BE719DFF4B22
5.NF.3 Interpret a fraction as division of the numerator by the denominator (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢 ÷ 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. - 26427339-66eb-49f4-9c63-4dab140e9c1b
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
6.F use data from a random sample to make inferences about a population; - 9F07172E-0D0A-11E2-9583-8B2E9DFF4B22
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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥² + 9𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥 + 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(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥 – 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦)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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦. - 753f388e-e7b0-4192-b79d-4dac6f0ca194
7.D generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties. - 9EEB17D6-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.NS.2.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. - 6e0ec566-7996-45c9-b5cf-b64ef59704ea
7.NS.2.a Understand that the multiplicative inverse of a number is its reciprocal and their product is equal to one. - 54c95f3a-7689-4616-9bce-3028abb5bccc
CCSS.Math.Content.6.SP.B.5.a Reporting the number of observations. - 1E9EE9FA-7053-11DF-8EBF-BE719DFF4B22
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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲 are integers, then –(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲) = (–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱)/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱/(–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲). Interpret quotients of rational numbers by describing real-world contexts. - 5526b516-9ec3-4ede-8383-1bb5811b7875
12.C compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations. - 9F134DE6-0D0A-11E2-9583-8B2E9DFF4B22
5.C solve mathematical and real-world problems involving similar shape and scale drawings. - 9F03CFB0-0D0A-11E2-9583-8B2E9DFF4B22
CE.7.2 The student will solve practical problems involving operations with rational numbers. - 21BE9AF8-C723-11E6-8B17-EE6DBF03DF2F
7.DSP.5.f Understand that a probability closer to 0 indicates an unlikely chance event. - 133471a1-0948-456a-b7d2-5475ac196f5e
4.C determine the constant of proportionality (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥) within mathematical and real-world problems; - 9F0121B6-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.7.NS.1.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - e7623cd2-ab35-4226-aa58-f271d92baa57
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 (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦 – 2)/(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥 – 1)(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥 + 1), (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥 – 1)(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 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.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
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
7.EE.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. - C575CA72-96FF-11E0-9509-C03D9DFF4B22
4.C give examples of ratios as multiplicative comparisons of two quantities describing the same attribute; - 9EE27496-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.K12.MP.5 Use appropriate tools strategically. - 7fb94672-da71-44b5-950f-b57258b31b63
MAFS.7.RP.1 Analyze proportional relationships and use them to solve real-world and mathematical problems. - 8bc4c4c6-0b0f-4c3c-a418-a34db69fa64c
CCSS.Math.Content.7.NS.A.1.a Describe situations in which opposite quantities combine to make 0. - 1EC910FE-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
7.GM.3 Describe two-dimensional cross-sections of three-dimensional figures, specifically right rectangular prisms and right rectangular pyramids. - aeecac36-76f9-4d06-bdb9-ff614a393271
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.7.RP.1.2.c Represent proportional relationships by equations. - 5fc42b42-51df-4370-ab0b-e7a40c31958f
CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. - A6430F50-6F89-11DF-BAEE-EA329DFF4B22
MAFS.7.SP.2.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. - 6548bfe8-f289-404e-8314-a36b1838f9bc
CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣 associated with a ratio 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢:𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣 with 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣 ≠ 0, and use rate language in the context of a ratio relationship. - 1E249010-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.7.EE.B.4.b Solve word problems leading to inequalities of the form 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲 > 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳 or 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲 < 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. - 1EE45C9C-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. - 1EB42CFC-7053-11DF-8EBF-BE719DFF4B22
7.EEI.3 Extend previous understanding of Order of Operations to solve multi-step real-world and mathematical problems involving rational numbers. Include fraction bars as a grouping symbol. - 8c573d8d-d83c-43ab-b7d0-39cbd75bc172
7.NS.5 Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, percentages). Exclude the conversion of repeating decimal numbers to fractions. - 3b896032-6713-4eed-881c-4acf695a06be
CCSS.Math.Content.6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. - 1E394C58-7053-11DF-8EBF-BE719DFF4B22
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
MG.8.5 The student will use the relationships among pairs of angles that are vertical angles, adjacent angles, supplementary angles, and complementary angles to determine the measure of unknown angles. - 489FB182-C725-11E6-8A30-53E9CCC8CA83
MAFS.7.NS.1.1.b Understand 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲 as the number located a distance |𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲| from 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱, in the positive or negative direction depending on whether 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲 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
7.DSP.5.e Understand that a probability close to ½ indicates that a chance event is neither likely nor unlikely. - c4acf844-678c-4c93-9659-045ca61a1dea
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
MG.8.6.a solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids; and - 5AB326BA-C725-11E6-A759-DEE9CCC8CA83
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
2.C locate, compare, and order integers and rational numbers using a number line; - 9EDCE22E-0D0A-11E2-9583-8B2E9DFF4B22
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.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
7.NS.2 Extend prior knowledge of operations with positive rational numbers to multiply and to divide all rational numbers. - 16ffee84-72ee-4fbf-8ec0-ca865884ae34
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.Practice.MP1 Make sense of problems and persevere in solving them. - A630F7A2-6F89-11DF-BAEE-EA329DFF4B22
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 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱 and 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲 are integers, then –(𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲) = (–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱)/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱/(–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲). Interpret quotients of rational numbers by describing real-world contexts. - 1ED1FDF4-7053-11DF-8EBF-BE719DFF4B22
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.Practice.MP7 Look for and make use of structure. - A648672A-6F89-11DF-BAEE-EA329DFF4B22
MAFS.6.NS.1.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. - 3321d844-34f9-4166-85c8-17edc576b39b
2.A classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers; - 9EDBF850-0D0A-11E2-9583-8B2E9DFF4B22
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
CCSS.Math.Content.6.RP.A.3.b Solve unit rate problems including those involving unit pricing and constant speed. - 1E2D1942-7053-11DF-8EBF-BE719DFF4B22
6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. - C556F98A-96FF-11E0-9509-C03D9DFF4B22
7.GM.6.b Understand that the concepts of volume and surface area are applied to three-dimensional figures such as cubes, right rectangular prisms, and right triangular prisms. - d744096f-7a8a-44c0-8e16-9f272453e103
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
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. - C53F5F00-96FF-11E0-9509-C03D9DFF4B22
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
MG.7.5 The student will solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles. - 5C9933EA-C723-11E6-B2AA-456EBF03DF2F
MAFS.7.RP.1.2 Recognize and represent proportional relationships between quantities. - 7c33b320-8b15-4141-a09b-b2d10f2a3012
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
7.DSP.1.d Understand that random sampling is used to gather a representative sample and supports valid inferences about the population. - 564abfe6-fcc4-46b6-abbb-0e2a17a8bca4
3.A add, subtract, multiply, and divide rational numbers fluently; and - 9EFEC84E-0D0A-11E2-9583-8B2E9DFF4B22
CCSS.Math.Content.7.SP.A Use random sampling to draw inferences about a population. - 1EF4393C-7053-11DF-8EBF-BE719DFF4B22
3.C represent integer operations with concrete models and connect the actions with the models to standardized algorithms; - 9EDFAD10-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.7.EE.1.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 717b1a64-7df5-4071-936a-4f5c4bab72a3
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
CCSS.Math.Content.6.RP.A.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. - 1E31940E-7053-11DF-8EBF-BE719DFF4B22
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
CCSS.Math.Content.7.RP.A.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - 1EB93512-7053-11DF-8EBF-BE719DFF4B22
MAFS.7.RP.1.3 Use proportional relationships to solve multistep ratio and percent problems. - 0b75b187-9efa-41ab-88bd-610c6f4e76fc
MAFS.6.RP.1.2 Understand the concept of a unit rate 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢/𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣 associated with a ratio 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢:𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣 with 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣 ≠ 0, and use rate language in the context of a ratio relationship. - e7681408-4886-4df5-a56c-3ffcbb848061
PS.8.12.c compare and analyze two data sets using boxplots. - 0490BEAE-C726-11E6-B693-22EBCCC8CA83
7.NS.1.e Apply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to add and subtract rational numbers. - 0e5f0e37-0fcb-4683-afc6-3230389a76c4
7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. - C580B644-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers. - 1E4523FC-7053-11DF-8EBF-BE719DFF4B22
10.A write one-variable, two-step equations and inequalities to represent constraints or conditions within problems; - 9F0E5DAE-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.SP.2.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. - a66d95b6-d488-413f-9d7f-8c8d97f542dc
MAFS.7.NS.1.1.c Understand subtraction of rational numbers as adding the additive inverse, 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱 – 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱 + (–𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲). 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
7.NS.4 Understand and apply the concepts of comparing and ordering to rational numbers. - 9f0673ea-9fc2-4aad-9215-6161a4e2a721
MAFS.7.SP.2 Draw informal comparative inferences about two populations. - 24c21b2b-4dc1-48da-9e96-2348e07b6d6c
7.G.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. - C57AA7AE-96FF-11E0-9509-C03D9DFF4B22
MAFS.K12.MP.3 Construct viable arguments and critique the reasoning of others. - 0ab2c4e3-2b87-4dd9-a12b-353a486762fa
7.EEI.2 Recognize that algebraic expressions may have a variety of equivalent forms and determine an appropriate form for a given real-world situation. - acde3fb9-382c-4e7c-98db-75c9d6199b42
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.6.SP.A Develop understanding of statistical variability. - 1E8FD2DA-7053-11DF-8EBF-BE719DFF4B22
6.C represent a given situation using verbal descriptions, tables, graphs, and equations in the form 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦𝑘𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦𝑘𝑥 or 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦𝑘𝑥𝑦 = 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦𝑘𝑥𝑦𝑥 + 𝘹𝘺𝘳𝘳𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘹𝘺𝘳𝘳𝑥𝑥𝑥𝑦𝑥𝑦𝘝𝘭𝘸𝘩𝘝𝘣𝘩𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘢𝘣𝘢𝘣𝘢𝘣𝘢𝘣𝑥𝑥𝑥𝑦𝑥𝑦𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝑘𝑦𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘢𝘣𝘢𝘣𝘣𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘱𝘲𝘢𝘣𝘢𝘣𝘣𝘱𝘲𝘱𝘲𝑦𝑘𝑥𝑦𝑥𝑏. - 9EE8C724-0D0A-11E2-9583-8B2E9DFF4B22
7.DSP.1.c Understand that generalizations from a sample are valid only if the sample is representative of the population. - 59accfbf-1320-4674-bd9b-f5a06b479fb6
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.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
7.RP.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - C566EA98-96FF-11E0-9509-C03D9DFF4B22
7.GM.2.b Decide if the measurements determine a unique triangle, more than one triangle, or no triangle. - 84939749-9806-4db7-b44a-8f09c0466917
7.G.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. - C57CA3EC-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.6.SP.B.5.c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. - 1EA3F940-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Practice.MP6 Attend to precision. - A647016E-6F89-11DF-BAEE-EA329DFF4B22
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
NS.6.1 The student will represent relationships between quantities using ratios, and will use appropriate notations, such as a/b, a to b, and a:b. - 88D84A86-C66A-11E6-8191-39EACCC8CA83
MAFS.6.NS.3.6.a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. - 8c49263b-c671-4b8e-8d8b-3cf8cdc0a451
7.GM.6.d Use the formulas for area, volume, and surface area appropriately. - f8f7cac7-48d5-4b31-956f-a2caf74dc875
CCSS.Math.Content.7.NS.A.2.c Apply properties of operations as strategies to multiply and divide rational numbers. - 1ED3352A-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
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
9.B determine the circumference and area of circles; - 9F0C77AA-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.SP.2.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. - b2e80bd1-166f-470a-889a-cfd4582af5b3
MAFS.6.NS.2.2 Fluently divide multi-digit numbers using the standard algorithm. - 5b40d095-6c96-48e6-9ff4-8e5db4197220
5.F distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form 34𝑦= 34𝑦𝑘34𝑦𝑘𝑥 or 34𝑦𝑘𝑥𝑦 = 34𝑦𝑘𝑥𝑦𝑚34𝑦𝑘𝑥𝑦𝑚𝑥 + 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏, where 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏 ≠ 0; - 9F23E8D6-0D0A-11E2-9583-8B2E9DFF4B22
6.G solve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents; - 9F078812-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.7.EE.2.4.a Solve word problems leading to equations of the form 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹 + 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲 = 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳 and 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱(34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹 + 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲) = 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 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
MAFS.6.SP.2.5.a Reporting the number of observations. - c3a58735-2dd0-4744-9076-860065f28f35
7.NS.2.b Understand sign rules for multiplying rational numbers. - 2c744b0c-36df-4e93-a8bf-7ffadc0612e5
MAFS.6.SP.1 Develop understanding of statistical variability. - c82cdde6-0186-4a24-b01e-324e0ab1302d
PFA.7.12 The student will solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable. - 70C644E2-C724-11E6-8D80-E26FBF03DF2F
MAFS.6.NS.3 Apply and extend previous understandings of numbers to the system of rational numbers. - ce84cbd5-eac0-4b0b-bf6c-4311036c7d9f
MAFS.7.RP.1.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - a63f040b-c2a1-4f32-9dc1-7bea05ec3e35
7.DSP.7 Apply the concepts of theoretical and experimental probabilities for simple events. - f63fa30b-03d1-46ec-8c42-d8ff3fb70fdd
CCSS.Math.Content.7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. - 1EF5932C-7053-11DF-8EBF-BE719DFF4B22
7.G.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. - C57E0F5C-96FF-11E0-9509-C03D9DFF4B22
7.NS.1.d Demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference. - 983b1735-87eb-4d69-a1d5-93ad6beb9e88
MAFS.7.RP.1.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. - a1018270-8c5e-4ad7-8f48-a6e9596dfc5d
CCSS.Math.Content.7.EE.B.4.a Solve word problems leading to equations of the form 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹 + 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲 = 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳 and 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱(34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹 + 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲) = 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 34𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 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
MAFS.7.SP.1 Use random sampling to draw inferences about a population. - 04a1ae14-941f-4997-8df7-d6f35fe69c52
7.DSP.2 Draw inferences about a population by collecting multiple random samples of the same size to investigate variability in estimates of the characteristic of interest. - b579bf16-6c57-496b-bff7-85fa2a8b8b18
7.RP.2 Identify and model proportional relationships given multiple representations, including tables, graphs, equations, diagrams, verbal descriptions, and real-world situations. - 7a0d9101-d34e-45df-972f-5ca3634f9bf5
7.DSP.5.c Understand that the probability of a chance event is a number between 0 and 1. - c39108ed-3818-47c1-9a72-8d8e29cbccca
5.A generalize the critical attributes of similarity, including ratios within and between similar shapes; - 9F02E82A-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.NS.3.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. - 534c67e1-2323-499b-81eb-ad4fb617147b
7.G.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. - C57D5A76-96FF-11E0-9509-C03D9DFF4B22
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.6.G.1.4 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. - c5cf0d9d-fbbc-481f-8c2d-7daa5250437f
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
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
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
9.A write one-variable, one-step equations and inequalities to represent constraints or conditions within problems; - 9EEE9C58-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.NS.3.7 Understand ordering and absolute value of rational numbers. - a928c4a0-7ce2-40ac-b529-b03919c04efa
NS.6.2.a represent and determine equivalencies among fractions, mixed numbers, decimals, and percents; and - 99E828BE-C66A-11E6-B4E3-0E71BF03DF2F
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
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
7.RP.3 Solve real-world and mathematical problems involving ratios and percentages using proportional reasoning (e.g., multi-step dimensional analysis, percent increase/decrease, tax). - 54527f43-bf71-47c7-81c5-d7f03bed497e
7.NS.1 Extend prior knowledge of operations with positive rational numbers to add and to subtract all rational numbers and represent the sum or difference on a number line. - 75ad1343-18f0-41f3-8456-197cf4bb7512
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
7.NS.2.d Apply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to multiply and divide rational numbers. - d77bfea8-34da-47a6-a4b0-badf3afe0613
12.A compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads; - 9F12419E-0D0A-11E2-9583-8B2E9DFF4B22
9.B represent solutions for one-variable, one-step equations and inequalities on number lines; and - 9EEF1458-0D0A-11E2-9583-8B2E9DFF4B22
9.A solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids; - 9F0BF406-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.SP.2.5 Summarize numerical data sets in relation to their context, such as by: - d5388c0c-2589-43b6-ad5e-739cd6c3e753
7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. - C56911EC-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm. - 1E3EC138-7053-11DF-8EBF-BE719DFF4B22
7.C determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations; and - 9EEA9E6E-0D0A-11E2-9583-8B2E9DFF4B22
7.DSP.1 Investigate concepts of random sampling. - 7ba2e596-f86d-4a92-8946-50f3c200a879
4.B calculate unit rates from rates in mathematical and real-world problems; - 9F00B30C-0D0A-11E2-9583-8B2E9DFF4B22
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
7.EEI.1 Apply mathematical properties (e.g., commutative, associative, distributive) to simplify and to factor linear algebraic expressions with rational coefficients. - 2f3d07af-afa9-405a-bec8-d6247aebd97a
MAFS.K12.MP.7 Look for and make use of structure. - 4b6d6fc7-f9ff-4e3c-89c9-3642b61650a0
4.D solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems; and - 9F01920E-0D0A-11E2-9583-8B2E9DFF4B22
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
7.NS.2.e Understand that some rational numbers can be written as integers and all rational numbers can be written as fractions or decimal numbers that terminate or repeat. - 47eb7df9-1575-4d14-a329-36b915c85f5e
MAFS.6.RP.1.3.b Solve unit rate problems including those involving unit pricing and constant speed. - ea97e4cf-2d7b-4d15-a870-05b5f5e3562d
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 (34𝘱𝘲𝘲𝘱𝘲𝑦 – 2)/(34𝘱𝘲𝘲𝘱𝘲𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥 + 1), (34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥² + 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 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
7.EEI.4.c Write and solve two-step linear inequalities. Graph the solution set on a number line and interpret its meaning. - 4d6a6f19-fc03-4271-8ef8-276e0962d281
7.GM.5 Write equations to solve problems involving the relationships between angles formed by two intersecting lines, including supplementary, complementary, vertical, and adjacent. - 2a29710b-7256-42f4-b0d1-a469a1ab8af6
MAFS.7.SP.3 Investigate chance processes and develop, use, and evaluate probability models. - 42650adb-825c-478c-8398-ffc653c7f412
CE.7.3 The student will solve single-step and multistep practical problems, using proportional reasoning. - 2C70F1D0-C723-11E6-A213-79E7CCC8CA83
MAFS.K12.MP.1 Make sense of problems and persevere in solving them. - efdcc5ef-fe77-460b-b67c-303a7528d39d
2.E extend representations for division to include fraction notation such as 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑎/34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑎𝑏 represents the same number as 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑎𝑏𝑎 ÷ 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑎𝑏𝑎𝑏 where 34𝘱𝘲𝘲𝘱𝘲𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑎𝑏𝑎𝑏𝑏 ≠ 0. - 9EDDDB84-0D0A-11E2-9583-8B2E9DFF4B22
7.DSP.5.a Determine probabilities of simple events. - c96989b4-1354-449c-a49d-54e0bcf3f50e
CCSS.Math.Content.6.NS.C.6.a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. - 1E4A881A-7053-11DF-8EBF-BE719DFF4B22
7.DSP.4 Compare the numerical measures of center (mean, median, mode) and variability (range, interquartile range, mean absolute deviation) from two random samples to draw inferences about the populations. - 5b0ae032-2e97-45f7-ae17-46111462a003
7.G.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. - C57B5A0A-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. - 1E629B3A-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.7.SP.C.8.c Design and use a simulation to generate frequencies for compound events. - 1F14547E-7053-11DF-8EBF-BE719DFF4B22
7.NS.1.b Understand that the sum of two rational numbers (p + q) represents a distance from p on the number line equal to |q| where the direction is indicated by the sign of q. - c2eed4c8-f6dd-4d66-b19f-02def620b96b
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
MAFS.6.SP.2.5.d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. - e15a29db-d7a3-4347-896a-f990073a48b1
7.NS.2.c Apply properties of operations as strategies to multiply and divide rational numbers. - C56F96DE-96FF-11E0-9509-C03D9DFF4B22
7.RP.2.d Use equations to model proportional relationships. - 3029f1e0-8b30-4642-bf1a-c34bc84b4347
CCSS.Math.Content.7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. - 1EF69826-7053-11DF-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
7.RP.1 Compute unit rates, including those involving complex fractions, with like or different units. - f923ad99-e551-4184-a5ca-29da4d234272
12.B use data from a random sample to make inferences about a population; and - 9F12BC1E-0D0A-11E2-9583-8B2E9DFF4B22
7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. - C5841FF0-96FF-11E0-9509-C03D9DFF4B22
7.EEI.4.b Write and solve multi-step linear equations that include the use of the distributive property and combining like ter