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Product Name: Reveal Math Algebra 1 2020
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Curriculum Standards:


PS.6.10.c compare circle graphs with the same data represented in bar graphs, pictographs, and line plots. - 368E124A-C66C-11E6-BD6F-A972BF03DF2F
CCSS.Math.Content.8.EE.C.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form 𝘹 = 𝘹𝘢, 𝘹𝘢𝘢 = 𝘹𝘢𝘢𝘢, or 𝘹𝘢𝘢𝘢𝘢 = 𝘹𝘢𝘢𝘢𝘢𝘣 results (where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣 are different numbers). - 1F399D6A-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.F-LE.1.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. - 614311d1-6d4c-4877-8d77-77eb6e55c895
CCSS.Math.Content.HSN-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. - 1F7642B0-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.A-SSE.2.3.a Factor a quadratic expression to reveal the zeros of the function it defines. - 6e8d693f-acb0-4a9c-a13e-5b0e76b6634c
CCSS.Math.Content.8.F.A Define, evaluate, and compare functions. - 1F461EC8-7053-11DF-8EBF-BE719DFF4B22
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
CCSS.Math.Content.HSA-SSE.B.3.a Factor a quadratic expression to reveal the zeros of the function it defines. - 1FCF76A0-7053-11DF-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
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. - C550C182-96FF-11E0-9509-C03D9DFF4B22
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
MAFS.8.SP.1 Investigate patterns of association in bivariate data. - 6fc197ec-34df-46f6-8b8b-08338bfeb315
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.6.EE.3.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. - 190e1387-e54a-48f2-b8e1-0066e9cf0827
12.C summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution; and - 9EF340C8-0D0A-11E2-9583-8B2E9DFF4B22
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
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
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.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
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
MAFS.6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. - 50458c9e-ed0f-4cba-a28a-f4957022f614
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
CCSS.Math.Content.6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents. - 1E5FE70A-7053-11DF-8EBF-BE719DFF4B22
MAFS.6.NS.2 Compute fluently with multi-digit numbers and find common factors and multiples. - c1801500-4a96-4f5a-88fc-259110d5d691
8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. - C5AF996E-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.HSA-SSE.B.3.c Use the properties of exponents to transform expressions for exponential functions. - 1FD28A34-7053-11DF-8EBF-BE719DFF4B22
MAFS.8.EE.2 Understand the connections between proportional relationships, lines, and linear equations. - 8251cb35-e9fa-496d-9b5b-dfb43e74b135
CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning. - A6497ED0-6F89-11DF-BAEE-EA329DFF4B22
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.7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems. - 1EB29FEA-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
CCSS.Math.Content.HSF-LE.A.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. - 20641210-7053-11DF-8EBF-BE719DFF4B22
MAFS.8.SP.1.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. - 4c4c1bea-b551-4e6e-938c-f450b83dde84
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, 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱 – 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱 + (–𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲). 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
MAFS.912.N-Q.1.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. - 4641014a-4a04-42ea-a758-242a7c410bc5
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.8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. - 1F1FA9D2-7053-11DF-8EBF-BE719DFF4B22
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
NS.8.3.a estimate and determine the two consecutive integers between which a square root lies; and - 1CC944A6-C725-11E6-8FA1-C9E9CCC8CA83
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 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲 > 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳 or 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲 < 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 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
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
CCSS.Math.Content.8.EE.C.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 1F3AE526-7053-11DF-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
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
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.8.EE.1 Work with radicals and integer exponents. - 56c44c41-66d7-4862-aa7a-4f89eb3d2fe2
PS.6.11 The student will - 41B88CE0-C66C-11E6-A13C-9372BF03DF2F
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
8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. - C592039A-96FF-11E0-9509-C03D9DFF4B22
12.A represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots; - 9EF25640-0D0A-11E2-9583-8B2E9DFF4B22
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.912.A-APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. - 789fc7d6-8cae-495f-9e63-c14fb1c4dfd6
MAFS.8.F.1 Define, evaluate, and compare functions. - e9818474-08d8-4863-b430-d242d74bd34b
CCSS.Math.Content.8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. - 1F368576-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.S-ID.2.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. - 0dea4bea-7abf-4711-8598-d7d8a3c941f9
PS.7.9.b make observations and inferences about data represented in a histogram; and - F607D798-C723-11E6-92A6-09E8CCC8CA83
CCSS.Math.Content.8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. - 1F3C4218-7053-11DF-8EBF-BE719DFF4B22
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
CCSS.Math.Content.HSA-REI.B.4.b Solve quadratic equations by inspection (e.g., for 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢 ± 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪 for real numbers 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣. - 200DF4F2-7053-11DF-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
5.C contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation; - 9F228B3A-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.HSF-BF.B.4.a Solve an equation of the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹) = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤 for a simple function 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. - 20552DCC-7053-11DF-8EBF-BE719DFF4B22
PS.7.9.a represent data in a histogram; - EF644552-C723-11E6-9ED5-4F6FBF03DF2F
MAFS.912.A-SSE.1.1.a Interpret parts of an expression, such as terms, factors, and coefficients. - 8f350077-a0f8-40fa-99b2-7f4e0d02815a
CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. - A6430F50-6F89-11DF-BAEE-EA329DFF4B22
CCSS.Math.Content.HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - 2010D992-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.F-LE.1.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. - 5bd8b229-eab1-4a8f-ae0b-d54e832a0ed8
CCSS.Math.Content.7.EE.B.4.b Solve word problems leading to inequalities of the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲 > 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳 or 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲 < 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳, where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱, 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲, and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳 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.HSS-ID.C.9 Distinguish between correlation and causation. - 21280AA8-7053-11DF-8EBF-BE719DFF4B22
2.B approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line; - 9F1BE3DE-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.912.S-ID.1.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). - 5c4a6eb4-8864-4748-a2b3-9f905af11859
MAFS.7.NS.1.1.b Understand 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲 as the number located a distance |𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲| from 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱, in the positive or negative direction depending on whether 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲 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
CCSS.Math.Content.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. - 1FFDCE7E-7053-11DF-8EBF-BE719DFF4B22
6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers. - C55130B8-96FF-11E0-9509-C03D9DFF4B22
8.EE.8 Analyze and solve pairs of simultaneous linear equations. - C59CF138-96FF-11E0-9509-C03D9DFF4B22
5.B represent linear non-proportional situations with tables, graphs, and equations in the form of 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥+𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏, where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏 ≠ 0; - 9F221290-0D0A-11E2-9583-8B2E9DFF4B22
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.8.NS.1.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. - 13d3d2c0-957d-4864-a633-dcbac4d39315
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
6.SP.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. - C563796C-96FF-11E0-9509-C03D9DFF4B22
8.SP.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. - C5B095E4-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. - A630F7A2-6F89-11DF-BAEE-EA329DFF4B22
CE.4.5.a determine common multiples and factors, including least common multiple and greatest common factor; - B537FF92-C65E-11E6-A731-9FD9CCC8CA83
CCSS.Math.Content.HSA-REI.B.4.a Use the method of completing the square to transform any quadratic equation in 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹 into an equation of the form (𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹 – 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱)² = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲 that has the same solutions. Derive the quadratic formula from this form. - 200CA278-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Practice.MP7 Look for and make use of structure. - A648672A-6F89-11DF-BAEE-EA329DFF4B22
8.F.3 Interpret the equation 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣 as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. - C5A2FE5C-96FF-11E0-9509-C03D9DFF4B22
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
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.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. - 202B0812-7053-11DF-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
NS.6.4 The student will recognize and represent patterns with whole number exponents and perfect squares. - D8841C54-C66A-11E6-887E-A2EACCC8CA83
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.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. - 1E5C1346-7053-11DF-8EBF-BE719DFF4B22
PS.8.12.c compare and analyze two data sets using boxplots. - 0490BEAE-C726-11E6-B693-22EBCCC8CA83
CCSS.Math.Content.6.RP.A.3.a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. - 1E2AE88E-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.F-BF.1.1.b Combine standard function types using arithmetic operations. - 2b53cd9d-2997-422d-90f1-057cd438db7a
7.NS.1.a Describe situations in which opposite quantities combine to make 0. - C56BA542-96FF-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
CCSS.Math.Content.HSF-IF.C.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. - 203A6D70-7053-11DF-8EBF-BE719DFF4B22
MAFS.7.NS.1.1.c Understand subtraction of rational numbers as adding the additive inverse, 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱 – 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱 + (–𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲). 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
MAFS.8.EE.2.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. - 98d51b29-de67-4abe-b7a6-32150edd2e62
MAFS.912.S-ID.3.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - ea4e177b-0dc6-4d90-bad7-6398ddb60b27
MAFS.7.SP.2 Draw informal comparative inferences about two populations. - 24c21b2b-4dc1-48da-9e96-2348e07b6d6c
PFA.8.16.e make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs. - A2B6B5AC-C726-11E6-8995-A872BF03DF2F
CCSS.Math.Content.HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - 2124F836-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.6.SP.A Develop understanding of statistical variability. - 1E8FD2DA-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
2.C convert between standard decimal notation and scientific notation; and - 9F1C5BDE-0D0A-11E2-9583-8B2E9DFF4B22
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
CCSS.Math.Content.HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. - 20606F34-7053-11DF-8EBF-BE719DFF4B22
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.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
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
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
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
CCSS.Math.Content.6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. - 1E7F5A18-7053-11DF-8EBF-BE719DFF4B22
CCSS.Math.Content.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context. - 1FC6D040-7053-11DF-8EBF-BE719DFF4B22
MAFS.8.G.1 Understand congruence and similarity using physical models, transparencies, or geometry software. - d20d06e9-6442-49a0-8f16-bd94d3cc66b0
MAFS.6.SP.1 Develop understanding of statistical variability. - c82cdde6-0186-4a24-b01e-324e0ab1302d
MAFS.6.NS.3 Apply and extend previous understandings of numbers to the system of rational numbers. - ce84cbd5-eac0-4b0b-bf6c-4311036c7d9f
MAFS.912.N-Q.1.2 Define appropriate quantities for the purpose of descriptive modeling. - ea155e2f-c077-4a91-b6c9-3415d50c5e5b
13.A interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots; and - 9EF4C330-0D0A-11E2-9583-8B2E9DFF4B22
CCSS.Math.Content.HSA-REI.B.4 Solve quadratic equations in one variable. - 200B3AAA-7053-11DF-8EBF-BE719DFF4B22
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.HSF-LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. - 2061D40A-7053-11DF-8EBF-BE719DFF4B22
12.D summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution. - 9EF3B81E-0D0A-11E2-9583-8B2E9DFF4B22
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
MAFS.912.A-REI.3.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. - 8ff0f22d-117f-49bc-a3b3-c068e87e737c
CCSS.Math.Content.HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. - 211C3D22-7053-11DF-8EBF-BE719DFF4B22
11.A construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data; - 9F2F9014-0D0A-11E2-9583-8B2E9DFF4B22
6.NS.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. - C54CF3A4-96FF-11E0-9509-C03D9DFF4B22
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.6.NS.3.7 Understand ordering and absolute value of rational numbers. - a928c4a0-7ce2-40ac-b529-b03919c04efa
CCSS.Math.Content.HSA-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. - 1FC38598-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.S-ID.2.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. - 75d55b82-83c7-43ca-ab1f-998a20e958f4
CCSS.Math.Content.HSF-IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - 20334950-7053-11DF-8EBF-BE719DFF4B22
MG.8.9.a verify the Pythagorean Theorem; and - A9BB3C20-C725-11E6-A8C2-9EEACCC8CA83
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
PFA.8.15.b determine the domain and range of a function. - 6A8DB3F6-C726-11E6-B100-5B72BF03DF2F
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
MAFS.8.EE.3.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢, 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢, or 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣 results (where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣 are different numbers). - 0e595aa8-da99-42e2-8e1c-d4052b3c7547
CCSS.Math.Content.8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. - 1F4FD706-7053-11DF-8EBF-BE719DFF4B22
8.C model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants; and - 9F2B5F62-0D0A-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
CCSS.Math.Content.HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). - 2065846A-7053-11DF-8EBF-BE719DFF4B22
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
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
6.SP.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. - C55F7A60-96FF-11E0-9509-C03D9DFF4B22
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 (𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦 – 2)/(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥 – 1)(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥 + 1), (𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥 – 1)(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 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
CE.7.3 The student will solve single-step and multistep practical problems, using proportional reasoning. - 2C70F1D0-C723-11E6-A213-79E7CCC8CA83
MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - caec55d0-18be-4e92-a8b1-8d8a3677f8b6
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
7.NS.1.b Understand 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲 as the number located a distance |𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲| from 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱, in the positive or negative direction depending on whether 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲 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. - C56C88FE-96FF-11E0-9509-C03D9DFF4B22
MAFS.912.F-IF.3.8.b Use the properties of exponents to interpret expressions for exponential functions. - 64d18729-b3c5-42f4-9ec3-d505f13a292c
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.HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. - 1F819E58-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.F-IF.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧 is a function and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹 is an element of its domain, then 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹) denotes the output of 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹. The graph of 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧 is the graph of the equation 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹). - 3b9e7038-0e8e-4ad9-b588-f99932d41bed
9 The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦 that simultaneously satisfy two linear equations in the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏 from the intersections of the graphed equations. - 9F2C4FA8-0D0A-11E2-9583-8B2E9DFF4B22
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
5.H identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems; and - 9F24D278-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.8.EE.3.8 Analyze and solve pairs of simultaneous linear equations. - 40f59586-e0b6-471a-8f8a-602a2a24be37
MAFS.912.A-SSE.2.3.c Use the properties of exponents to transform expressions for exponential functions. - cdb48175-78dd-4251-88e8-121f0ab727a6
MAFS.912.S-ID.1.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - ef092516-4307-497b-8ffb-f57bfe6d5100
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
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.8.EE.1.2 Use square root and cube root symbols to represent solutions to equations of the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹² = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹³ = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱, where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱 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
CCSS.Math.Content.6.SP.B Summarize and describe distributions. - 1E99E8E2-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.A-CED.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. - 18186a04-0947-4402-bdac-f6aba4c9e901
CCSS.Math.Content.8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. - 1F6B252E-7053-11DF-8EBF-BE719DFF4B22
12.B use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution; - 9EF2D098-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.6.EE.2 Reason about and solve one-variable equations and inequalities. - a971566c-0f10-4450-8b7c-8c0e1e160ab5
MAFS.7.RP.1.2.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. - f1ba40b5-3f95-4218-88d0-a8f81e6dbfb2
CCSS.Math.Content.8.NS.A Know that there are numbers that are not rational, and approximate them by rational numbers. - 1F1E4754-7053-11DF-8EBF-BE719DFF4B22
8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. - C5AEABBC-96FF-11E0-9509-C03D9DFF4B22
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.K12.MP.8 Look for and express regularity in repeated reasoning. - 6ff0dba2-4ea2-454f-8c62-e326c5d5841a
MAFS.6.SP.1.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. - 863dde88-f7b8-4db6-b4be-3f2dffd27a2f
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.6.NS.C.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. - 1E46DA08-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.S-ID.3.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. - f30051bf-3298-4ca0-90c8-d33d533a9bfc
NS.6.3.c identify and describe absolute value of integers. - CA865BDA-C66A-11E6-99A8-58EACCC8CA83
6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. - C5502D58-96FF-11E0-9509-C03D9DFF4B22
CCSS.Math.Content.HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - 2023EA78-7053-11DF-8EBF-BE719DFF4B22
PS.8.13.a represent data in scatterplots; - 16395A3A-C726-11E6-8878-23EBCCC8CA83
6.SP.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. - C5630B3A-96FF-11E0-9509-C03D9DFF4B22
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹² = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱 and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹³ = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱, where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - C5935A42-96FF-11E0-9509-C03D9DFF4B22
MAFS.912.A-REI.4.11 Explain why the 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹-coordinates of the points where the graphs of the equations 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹) and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - 8747c55b-cbc9-4ef6-b38a-95fde44f15c3
CCSS.Math.Practice.MP4 Model with mathematics. - A6446404-6F89-11DF-BAEE-EA329DFF4B22
MAFS.912.A-REI.4.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). - a764f529-16f8-4fa9-b5cd-3d0831349fe3
CCSS.Math.Content.HSA-REI.D.11 Explain why the 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹-coordinates of the points where the graphs of the equations 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹) and 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - 201C0FBA-7053-11DF-8EBF-BE719DFF4B22
8.EE.8.c Solve real-world and mathematical problems leading to two linear equations in two variables. - C59FABA8-96FF-11E0-9509-C03D9DFF4B22
MG.8.9.b apply the Pythagorean Theorem. - B1231C9E-C725-11E6-A4B9-9AEACCC8CA83
MAFS.6.SP.1.2 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. - 3b2fab55-8d5b-4046-b228-4a2b1c1c8111
8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. - C5A4E03C-96FF-11E0-9509-C03D9DFF4B22
PS.8.12.b make observations and inferences about data represented in boxplots; and - FBAB8A12-C725-11E6-AC99-C571BF03DF2F
8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. - C5A18C02-96FF-11E0-9509-C03D9DFF4B22
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
7.C use the Pythagorean Theorem and its converse to solve problems; and - 9F28F664-0D0A-11E2-9583-8B2E9DFF4B22
6.SP.5 Summarize numerical data sets in relation to their context, such as by: - C5614AB6-96FF-11E0-9509-C03D9DFF4B22
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.8.EE.3.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 549f7952-95bd-49ac-8d89-bbb9f4152ef1
7.3 The student will identify and apply the following properties of operations with real numbers: the commutative and associative properties for addition and multiplication; the distributive property; the additive and multiplicative identity properties; the additive and multiplicative inverse properties; and the multiplicative property of zero. - A8BF6708-0183-11D9-B5BC-D75BD0522FDF
PS.6.11.a represent the mean of a data set graphically as the balance point; and - 4ACECB82-C66C-11E6-AB88-94EBCCC8CA83
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.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. - fdb954fb-31eb-437a-9842-26c4438291f5
11.B determine if the given value(s) make(s) one-variable, two-step equations and inequalities true; and - 9F10CB16-0D0A-11E2-9583-8B2E9DFF4B22
CCSS.Math.Content.HSA-CED.A.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. - 1FF95682-7053-11DF-8EBF-BE719DFF4B22
MAFS.912.A-APR.3.4 Prove polynomial identities and use them to describe numerical relationships. - f414f786-5e26-45a2-a1c4-330764ca685d
MAFS.8.EE.3.8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. - 5f945ec3-6a31-433e-acdb-d27309491694
MAFS.8.F.2 Use functions to model relationships between quantities. - 9272ab99-4d9c-41cc-8e71-78f3c1df8b7a
MAFS.8.EE.2.6 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 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹 for a line through the origin and the equation 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘺 = 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘺𝘮𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘺𝘮𝘹 + 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘺𝘮𝘹𝘣 for a line intercepting the vertical axis at 𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝘱𝘲𝘱𝘲𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘹𝘢𝘣𝘪𝘢𝘣𝘧𝘹𝘤𝘧𝘱𝘹𝘲𝘳𝘱𝘹𝘲𝘳𝘱𝘲𝘳𝘱𝘲𝘲𝘱𝘲𝑦𝑚𝑥𝑏𝑏𝘹𝘹𝘱𝘲𝘺𝘮𝘹𝘣𝘱𝘲𝘱𝘲𝘹𝘢𝘢𝘢𝘢𝘣𝘢𝘣𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘱𝘲𝘲𝘱𝘲𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹𝑥𝑦𝑦𝑚𝑥𝑏𝘹𝘱𝘹𝘱𝘱𝘹𝘱𝘹𝘱𝘱𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘺𝘮𝘹𝘺𝘮𝘹𝘣𝘣. - 95d89ffb-9b92-4e37-87b8-8b28f88d7132
MAFS.8.NS.1.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). - 2f3b4c16-4ad7-4def-9c63-0ac8ab7039f3
CCSS.Math.Content.HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. - 211DCEA8-7053-11DF-8EBF-BE719DFF4B22
MAFS.K12.MP.5.1 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. - 85786a83-8e8e-4f2c-9ee5-c473a733533a
CCSS.Math.Content.HSF-BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context. - 20476DA4-7053-11DF-8EBF-BE719DFF4B22
5.D use a trend line that approximates the linear relationship between bivariate sets of data to make predictions; - 9F22FC8C-0D0A-11E2-9583-8B2E9DFF4B22
MAFS.912.N-RN.1.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - b8a7e69e-d432-491e-a178-6d0c6369ece1
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
NS.8.3.b determine both the positive and negative square roots of a given perfect square. - 24016C26-C725-11E6-8FA1-C9E9CCC8CA83
MAFS.6.SP.2.5.c Giving quantitative measures of center (median and/or