Organization: SAVVAS Product Name: Mississippi enVisionmath2.0 Grades 6-8 Grade 8 Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-f427b27b-ecc4-3e03-ac0d-6b5a2497c96a Timestamp: Tuesday, December 29, 2020 03:13 PM EST Status: VALID! Conformant: true ----- VALID! ----- Resource Validation Results The document is valid. ----- VALID! ----- Schema Location Results Schema locations are valid. ----- VALID! ----- Schema Validation Results The document is valid. ----- VALID! ----- Schematron Validation Results The document is valid. Curriculum Standards: 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. - 8.NS.1 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). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - 8.NS.2 Use the Converse of the Pythagorean Theorem to identify right triangles. - 8.G.7.2 Use the Pythagorean Theorem to solve problems. - 8.G.7.3 Use the Pythagorean Theorem to find unknown sides of triangles. - 8.G.7.1 Assess the reasonableness of predictions using scatterplots by interpreting them in the original context. - 8.4.1.3 Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit and determine an equation for the line. Use appropriate titles, labels and units. Know how to use graphing technology to display scatterplots and corresponding lines of best fit. - 8.4.1.1 Use a line of best fit to make statements about approximate rate of change and to make predictions about values not in the original data set. - 8.4.1.2 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Express answers in scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology (e.g., interpret 4.7EE9 displayed on a calculator as 4.7 x 10^9). - M08.B-E.1.1.4 Estimate very large or very small quantities by using numbers expressed in the form of a single digit times an integer power of 10 and express how many times larger or smaller one number is than another. - M08.B-E.1.1.3 Use square root and cube root symbols to represent solutions to equations of the form x^2 – p and x^3 – p, where p is a positive rational number. Evaluate square roots of perfect squares (up to and including 12^2) and cube roots of perfect cubes (up to and including 5^3) without a calculator. - M08.B-E.1.1.2 Apply one or more properties of integer exponents to generate equivalent numerical expressions without a calculator (with final answers expressed in exponential form with positive exponents). Properties will be provided. - M08.B-E.1.1.1 The student will estimate and determine the two consecutive integers between which a square root lies. - 8.3a The student will determine both the positive and negative square roots of a given perfect square. - 8.3b Mathematical Modeling: Congruence and Similarity - 8.G.6MM Investigate if orientation is preserved under rigid transformations. - 8.GM.A.1b Find the relative frequencies of two-way tables and interpret what they mean. - 8.DP.4.5 Display and interpret relationships between paired categorical data. - 8.DP.4.4 Make a prediction by using the equation of a line that closely fits a set of data. - 8.DP.4.3 Use a line to represent the relationship between paired data. - 8.DP.4.2 Construct a scatter plot and use it to understand the relationship between paired data. - 8.DP.4.1 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - MAFS.K12.MP.3.1 Use the Pythagorean Theorem to find the distance between two points in the coordinate plane. - 8.G.7.4 Verify that angle measure, betweeness, collinearity and distance are preserved under rigid transformations. - 8.GM.A.1a Find the surface areas of cylinders, cones, and spheres. - 8.G.8.1 Recognize functions given a table of values or a set of ordered pairs. - NC.8.F.1.b Use what I know about finding volumes of rectangular prisms to find the volume of a cylinder. - 8.G.8.2 Recognize functions when graphed as the set of ordered pairs consisting of an input and exactly one corresponding output. - NC.8.F.1.a Lines are taken to lines, and line segments to line segments of the same length. - MAFS.8.G.1.1a Angles are taken to angles of the same measure. - MAFS.8.G.1.1b Know and apply the properties of integer exponents to generate equivalent numerical expressions. - NY-8.EE.1 Parallel lines are taken to parallel lines. - MAFS.8.G.1.1c Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology. - NY-8.EE.4 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. - NY-8.EE.5 Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational. - NY-8.EE.2 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. - NY-8.EE.3 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions. - NY-8.NS.2 Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational. - NY-8.NS.1 Analyze and solve pairs of simultaneous linear equations. - 8.EE.8 Mathematical Modeling: Understand and Apply the Pythagorean Theorem - 8.G.7MM Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27. - 8.EE.1 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 8.EE.2 Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. - M.8.10a Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10⁸ and the population of the world as 7 times 10⁹, and determine that the world population is more than 20 times larger. - 8.EE.3 Solve real-world and mathematical problems leading to two linear equations in two variables. (e.g., Given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.) - M.8.10c Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology - 8.EE.4 Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection. (e.g., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.) - M.8.10b Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.5 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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.6 Analyze and solve pairs of simultaneous linear equations. - NY-8.EE.8 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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - NY-8.EE.6 Solve linear equations in one variable. - NY-8.EE.7 Reason abstractly and quantitatively. 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. - MAFS.K12.MP.2.1 Find the volumes of cones. - 8.G.8.3 Find the volume of a sphere and use it to solve problems. - 8.G.8.4 The student will solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids. - 8.6a Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, percentages). Include the conversion of repeating decimal numbers to fractions. - 8.NS.3 Given two congruent figures, describe a sequence that exhibits the congruence between them. - NC.8.G.2.c Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. - NC.8.G.2.b Solve systems of equations using elimination. - 8.AF.5.4 Evaluate square roots of perfect squares and cube roots of perfect cubes for positive numbers less than or equal to 400. - NC.8.EE.2.b Represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. - NC.8.EE.2.a Verify experimentally the properties of rotations, reflections, and translations that create congruent figures. - NC.8.G.2.a Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of the slope and y-intercept of its graph or a table of values. - NC.8.F.4.d Use informal arguments to analyze angle relationships. - NC.8.G.5 Describe a possible sequence of transformations between two similar figures. - 8.GM.A.4a Look for and express regularity in repeated reasoning. - 8.MP.8 Construct a graph of a linear relationship given an equation in slope-intercept form. - NC.8.F.4.c Use transformations to define similarity. - NC.8.G.4 Look for and make use of structure. - 8.MP.7 Write an equation in slope-intercept form to model a linear relationship by determining the rate of change and the initial value, given at least two (x, y) values or a graph. - NC.8.F.4.b The student will given a polygon, apply transformations, to include translations, reflections, and dilations, in the coordinate plane. - 8.7a Describe the effect of dilations about the origin, translations, rotations about the origin in 90 degree increments, and reflections across the x-axis and y-axis on two-dimensional figures using coordinates. - NC.8.G.3 Perform operations with numbers in scientific notation. - 8.AF.1.10 Attend to precision. - 8.MP.6 Understand that a linear relationship can be generalized by 𝑦 =mx + b. - NC.8.F.4.a The student will identify practical applications of transformations. - 8.7b Use transformations to define congruence. - NC.8.G.2 Use appropriate tools strategically. - 8.MP.5 Solve systems of equations using substitution. - 8.AF.5.3 Find the solution to a system of equations using graphs. - 8.AF.5.2 Find the number of solutions of a system of equations by inspecting the equations. - 8.AF.5.1 Mathematical Modeling: Solve Problems Involving Surface Area and Volume - 8.G.8MM Model with mathematics. - 8.MP.4 Understand how the formulas for the volumes of cones, cylinders, and spheres are related and use the relationship to solve real-world and mathematical problems. - NC.8.G.9 The student will verify the Pythagorean Theorem. - 8.9a Construct viable arguments and critique the reasoning of others. - 8.MP.3 The student will apply the Pythagorean Theorem. - 8.9b Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - NC.8.G.8 Reason abstractly and quantitatively. - 8.MP.2 Apply the Pythagorean Theorem and its converse to solve real-world and mathematical problems. - NC.8.G.7 Make sense of problems and persevere in solving them. - 8.MP.1 Explain the Pythagorean Theorem and its converse. - NC.8.G.6 Model the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers. - 8.NS.1c Understand that all real numbers have a decimal expansion. - 8.NS.1b Recognize the differences between rational and irrational numbers. - 8.NS.1a Use appropriate tools strategically. 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. - MAFS.K12.MP.5.1 Parallel lines are taken to parallel lines. - 8.G.A.1c Angles are taken to angles of the same measure. - 8.G.A.1b Lines are taken to lines, and line segments to line segments of the same length. - 8.G.A.1a Explore the relationship between the interior and exterior angles of a triangle. - 8.GM.A.5b Construct and explore the angles created when parallel lines are cut by a transversal. - 8.GM.A.5c Use the properties of similar figures to solve problems. - 8.GM.A.5d Derive the sum of the interior angles of a triangle. - 8.GM.A.5a Informally fit a straight line for a scatter plot that suggests a linear association. - NC.8.SP.2.a Informally assess the model fit by judging the closeness of the data points to the line. - NC.8.SP.2.b Qualitatively analyze the functional relationship between two quantities. - NC.8.F.5 Analyze functions that model linear relationships. - NC.8.F.4 Describe a possible sequence of rigid transformations between two congruent figures. - 8.GM.A.2a Identify linear functions from tables, equations, and graphs. - NC.8.F.3 Compare properties of two linear functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). - NC.8.F.2 Understand that a function is a rule that assigns to each input exactly one output. - NC.8.F.1 The student will make observations about data represented in scatterplots. - 8.13b The student will use a drawing to estimate the line of best fit for data represented in a scatterplot. - 8.13c The student will represent data in scatterplots. - 8.13a Analyze and solve a system of two linear equations in two variables in slope-intercept form. - NC.8.EE.8 Perform multiplication and division with numbers expressed in scientific notation to solve real-world problems, including problems where both decimal and scientific notation are used. - NC.8.EE.4 Use numbers expressed in scientific notation to estimate very large or very small quantities and to express how many times as much one is than the other. - NC.8.EE.3 Develop and apply the properties of integer exponents to generate equivalent numerical expressions. - NC.8.EE.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. - 8.NS.A.1 Estimate the value and compare the size of irrational numbers and approximate their locations on a number line. - 8.NS.A.2 Understand that every number has a decimal expansion. Building upon the definition of a rational number, know that an irrational number is defined as a non-repeating, non-terminating decimal. - NC.8.NS.1 Construct and interpret scatter plots of bivariate measurement data to investigate patterns of association between two quantities - 8.DSP.A.1 Interpret the parameters of a linear model of bivariate measurement data to solve problems. - 8.DSP.A.3 Generate and use a trend line for bivariate data, and informally assess the fit of the line. - 8.DSP.A.2 Model with mathematics. 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. - MAFS.K12.MP.4.1 Use rational approximations of irrational numbers to compare the size of irrational numbers and locate them approximately on a number line. Estimate the value of expressions involving: - NC.8.NS.2 Use angle measures to determine whether two triangles are similar. - 8.G.6.10 Recognize that square roots of non-perfect squares are irrational. - 8.EEI.2d Evaluate cube roots of perfect cubes. - 8.EEI.2c Evaluate square roots of perfect squares. - 8.EEI.2b Find the exact and approximate solutions to equations of the form 𝑦𝑥²=𝑦𝑥𝑝 and x³=𝑦𝑥𝑝𝑝 where 𝑦𝑥𝑝𝑝𝑝 is a positive rational number. - 8.EEI.2a Verify experimentally the properties of dilations that create similar figures. - NC.8.G.4.a Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - NC.8.G.4.c Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. - NC.8.G.4.b π to the hundredths. - NC.8.NS.2.b Solve real-world and mathematical problems leading to systems of linear equations by graphing the equations. Solve simple cases by inspection. - NC.8.EE.8.b Understand that solutions to a system of two linear equations correspond to the points of intersection of their graphs because the point of intersection satisfies both equations simultaneously. - NC.8.EE.8.a Square roots and cube roots to the tenths. - NC.8.NS.2.a Express very large and very small quantities in scientific notation in the form 𝑦𝑥𝑝𝑝𝑝𝑎×10ᵇ=𝑦𝑥𝑝𝑝𝑝𝑎𝑝 where 1≤𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎<10 and 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏 is an integer. - 8.EEI.3a Estimate and compare the relative size of two quantities in scientific notation. - 8.EEI.3c Translate between decimal notation and scientific notation. - 8.EEI.3b 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 = a, a = a, or a = b results (where a and b are different numbers). - 8.EE.7a Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 8.G.C.9 Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 8.EE.7b Draw an approximate line of best fit on a scatter plot that appears to have a linear association and informally assess the fit of the line to the data points. - 8.DSP.2 Look for and make use of structure. 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 x² + 9x + 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(x – y)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 x and y. - MAFS.K12.MP.7.1 Analyze and solve pairs of simultaneous linear equations. - 8.EE.C.8 Understand and apply the laws of exponents (i.e., product rule, quotient rule, power to a power, product to a power, quotient to a power, zero power property, negative exponents) to simplify numerical expressions that include integer exponents. - 8.EEI.1 Compare two different proportional relationships. - 8.EEI.B.5b Interpret the unit rate as the slope of the graph. - 8.EEI.B.5a Use relative frequencies calculated for rows or columns to describe possible association between the two variables. - NC.8.SP.4.b Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. - NC.8.SP.4.a Sketch a graph that exhibits the qualitative features of a real-world function. - NC.8.F.5.b Analyze a graph determining where the function is increasing or decreasing; linear or non-linear. - NC.8.F.5.a Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them. - M08.C-G.1.1.4 Use similar right triangles to show and explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. - M08.B-E.2.1.2 Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - M08.B-E.2.1.3 Use relative frequencies calculated for rows or columns to describe possible association between the two variables. - 8.DSP.A.4b Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. - M08.B-E.2.1.1 Explain why the slope (m) is the same between any two distinct points on a non-vertical line in the Cartesian coordinate plane. - 8.EEI.B.6a Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. - 8.DSP.A.4a Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EEI.B.6b Attend to precision. 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. - MAFS.K12.MP.6.1 Represent real-world situations using equations and inequalities involving one variable. - PA.A.4.3 Illustrate, write, and solve mathematical and real-world problems using linear equations with one variable with one solution, infinitely many solutions, or no solutions. Interpret solutions in the original context. - PA.A.4.1 Identify and apply properties of rotations, reflections, and translations. - M08.C-G.1.1.1 Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them. - M08.C-G.1.1.2 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - M08.C-G.1.1.3 Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real-world problems. - NY-8.G.9 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - NY-8.G.8 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - NY-8.G.7 Explain why solution(s) to a system of two linear equations in two variables correspond to point(s) of intersection of the graphs. - 8.EEI.C.8b Explain why systems of linear equations can have one solution, no solution or infinitely many solutions. - 8.EEI.C.8c Relate equations for proportional relationships (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥) with the slope-intercept form (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥+𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏) where 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏=0. - 8.EEI.6c Solve systems of two linear equations. - 8.EEI.C.8d Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve simple cases by inspection. - NY-8.EE.8b Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane. - NY-8.G.2 Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients. - NY-8.EE.8c Verify experimentally the properties of rotations, reflections, and translations. - NY-8.G.1 Derive the slope-intercept form (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥+𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏) for a non-vertical line. - 8.EEI.6b Explain why the slope, 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚, is the same between any two distinct points on a non-vertical line using similar triangles. - 8.EEI.6a Understand a proof of the Pythagorean Theorem and its converse. - NY-8.G.6 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. - NY-8.G.5 Mathematical Modeling: Analyze and Solve Systems of Linear Equations - 8.DP.5MM Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane. - NY-8.G.4 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - NY-8.G.3 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? - 8.SP.A.4 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. - 8.SP.A.3 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. - 8.SP.A.2 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. - 8.SP.A.1 Graph systems of linear equations and recognize the intersection as the solution to the system. - 8.EEI.C.8a Justify why linear equations have a specific type of solution. - 8.EEI.7d Solve linear equations and inequalities with rational number coefficients that include the use of the distributive property, combining like terms, and variables on both sides. - 8.EEI.7a Generate linear equations with the three types of solutions. - 8.EEI.7c Recognize the three types of solutions to linear equations: one solution (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎), infinitely many solutions (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎), or no solutions (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏). - 8.EEI.7b Calculate experimental probabilities and represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown. - PA.D.2.1 Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 8.EE.A.2 Lines are taken to lines, and line segments to line segments of the same length. - M.8.16a Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 × 3^-5 = 3^-3 = 1/33 = 1/27. - 8.EE.A.1 Parallel lines are taken to parallel lines. - M.8.16c Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology - 8.EE.A.4 Angles are taken to angles of the same measure. - M.8.16b Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10^8 and the population of the world as 7 times 10^9, and determine that the world population is more than 20 times larger. - 8.EE.A.3 Select appropriate units of measure when representing answers in scientific notation. - 8.EEI.4b Rotate a two-dimensional figure. - 8.G.6.3 Describe and perform a sequence of transformations. - 8.G.6.4 Multiply and divide numbers expressed in both decimal and scientific notation. - 8.EEI.4a Translate two-dimensional figures. - 8.G.6.1 Reflect two-dimensional figures. - 8.G.6.2 Translate how different technological devices display numbers in scientific notation. - 8.EEI.4c Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of functions that are linear and non-linear. - NY-8.F.3 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). - NY-8.F.2 Recognize the relationships between the angles created when parallel lines are cut by a transversal. - NC.8.G.5.b 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. - NY-8.F.1 Recognize relationships between interior and exterior angles of a triangle. - NC.8.G.5.a Solve real-world and mathematical problems involving angles. - NC.8.G.5.d Recognize the angle-angle criterion for similarity of triangles. - NC.8.G.5.c Describe qualitatively the functional relationship between two quantities by analyzing a graph.Sketch a graph that exhibits the qualitative features of a function that has been described in a real-world context. - NY-8.F.5 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - NY-8.F.4 Mathematical Modeling: Investigate Bivariate Data - 8.DP.4MM Recognize linear equations in one variable as having one solution, infinitely many solutions, or no solutions. - NC.8.EE.7.a Solve linear equations and inequalities including multi-step equations and inequalities with the same variable on both sides. - NC.8.EE.7.b Create and identify linear equations with one solution, infinitely many solutions or no solutions. - 8.EEI.C.7a Compare two different proportional relationships given multiple representations, including tables, graphs, equations, diagrams, and verbal descriptions. - 8.EEI.5c Solve linear equations and inequalities with rational number coefficients, including equations and inequalities whose solutions require expanding expressions using the distributive property and combining like terms. - 8.EEI.C.7b Interpret unit rate as the slope of the graph. - 8.EEI.5b Graph proportional relationships. - 8.EEI.5a Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit, make statements about average rate of change, and make predictions about values not in the original data set. Use appropriate titles, labels and units. - PA.D.1.3 Solve real-world and mathematical problems leading to two linear equations in two variables. - M08.B-E.3.1.5 Apply the Pythagorean theorem to find the distance between two points in a coordinate system. - M08.C-G.2.1.3 Write and identify 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 = a, a = a, or a = b results (where a and b are different numbers). - M08.B-E.3.1.1 Apply the converse of the Pythagorean theorem to show a triangle is a right triangle. - M08.C-G.2.1.1 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.B.5 Solve linear equations that have rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - M08.B-E.3.1.2 Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Figures provided for problems in three dimensions will be consistent with Eligible Content in grade 8 and below.) - M08.C-G.2.1.2 Interpret solutions to a system of two linear equations in two variables as points of intersection of their graphs because points of intersection satisfy both equations simultaneously. - M08.B-E.3.1.3 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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.B.6 Look for and express regularity in repeated reasoning. 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 (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 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. - MAFS.K12.MP.8.1 Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection. - M08.B-E.3.1.4 Use a sequence of transformations, including dilations, to show that figures are similar. - 8.G.6.7 Identify and find the measures of angles formed by parallel lines and a transversal. - 8.G.6.8 Use a sequence of translations, reflections, and rotations to show that figures are congruent. - 8.G.6.5 Dilate two-dimensional figures. - 8.G.6.6 Find the interior and exterior angle measures of a triangle. - 8.G.6.9 Solve equations of the form x² = p and x³ = p, where p is a positive rational number. - 8.EEI.A.2a Recognize that square roots of non-perfect squares are irrational. - 8.EEI.A.2c Evaluate square roots of perfect squares less than or equal to 625 and cube roots of perfect cubes less than or equal to 1000. - 8.EEI.A.2b 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 associations between the two variables. - M08.D-S.1.2.1 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. - 8.F.A.3 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. - 8.F.A.1 Compare characteristics of two functions each represented in a different way. - 8.F.A.2 Understand that all rational numbers have a decimal expansion that terminates or repeats. - 8.NS.A.1b Know the differences between rational and irrational numbers. - 8.NS.A.1a Generate equivalent representations of rational numbers. - 8.NS.A.1d Convert decimals which repeat into fractions and fractions into repeating decimals. - 8.NS.A.1c Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - MAFS.8.G.3.9 Determine the parameters of a linear function. - 8.F.B.4b Apply formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. Formulas will be provided. - M08.C-G.3.1.1 Determine the x-intercept of a linear function. - 8.F.B.4c Understand and verify that a solution to a system of linear equations is represented on a graph as the point of intersection of the two lines. - 8.EEI.8b Graph systems of linear equations and estimate their point of intersection. - 8.EEI.8a Understand that systems of linear equations can have one solution, no solution, or infinitely many solutions. - 8.EEI.8d Solve systems of linear equations algebraically, including methods of substitution and elimination, or through inspection. - 8.EEI.8c Explain the parameters of a linear function based on the context of a problem. - 8.F.B.4a Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading those from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values. - M08.B-F.2.1.1 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 or determine a graph that exhibits the qualitative features of a function that has been described verbally. - M08.B-F.2.1.2 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 correlation, linear association, and nonlinear association. - M08.D-S.1.1.1 For scatter plots that suggest a linear association, identify a line of best fit by judging the closeness of the data points to the line. - M08.D-S.1.1.2 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. - M08.D-S.1.1.3 Develop and apply the properties of integer exponents, including a⁰ = 1 (with a ≠ 0), to generate equivalent numerical and algebraic expressions. - PA.N.1.1 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 8.F.B.4 Describe the functional relationship between two quantities from a graph or a verbal description. - 8.F.B.5 Classify real numbers as rational or irrational. Explain why the rational number system is closed under addition and multiplication and why the irrational system is not. Explain why the sum of a rational number and an irrational number is irrational; and the product of a non-zero rational number and an irrational number is irrational. - PA.N.1.4 Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms. - NY-8.EE.7a Compare real numbers; locate real numbers on a number line. Identify the square root of a perfect square to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers. - PA.N.1.5 Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms. - NY-8.EE.7b Express and compare approximations of very large and very small numbers using scientific notation. - PA.N.1.2 Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation. - PA.N.1.3 Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. - 8.EE.C.8c Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. - 8.EE.C.8a Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. - 8.EE.C.8b Understand the concept of surface area and find surface area of pyramids. - 8.GM.C.9a Understand the concepts of volume and find the volume of pyramids, cones and spheres. - 8.GM.C.9b Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Recognize when the system has one solution, no solution, or infinitely many solutions. - NY-8.EE.8a Understand that a function assigns to each input exactly one output. - 8.F.A.1a Make sense of problems and persevere in solving them. - MP.1 Reason abstractly and quantitatively. - MP.2 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - MAFS.8.G.2.7 Explain a proof of the Pythagorean Theorem and its converse. - MAFS.8.G.2.6 Determine if a relation is a function. - 8.F.A.1b Graph a function. - 8.F.A.1c Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - MAFS.8.G.2.8 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. Instructional Note: A decimal expansion that repeats the digit 0 is often referred to as a “terminating decimal.” - M.8.1 Look for and make use of structure. - MP.7 Look for and express regularity in repeated reasoning. - MP.8 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 such as π2. (e.g., By truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.) - M.8.2 Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 8.EE.C.7b Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. (e.g., Estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.) - M.8.5 Construct viable arguments and critique the reasoning of others. - MP.3 Model with mathematics. - MP.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. (e.g., Use millimeters per year for seafloor spreading.) Interpret scientific notation that has been generated by technology. - M.8.6 Know and apply the properties of integer exponents to generate equivalent numerical expressions. (e.g., 32 × 3–5 = 3–3 = 1/33 = 1/27.) - M.8.3 Use appropriate tools strategically. - MP.5 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - M.8.4 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 = a, a = a, or a = b results (where a and b are different numbers). - 8.EE.C.7a Attend to precision. - MP.6 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. Instructional Note: Function notation is not required in grade 8. - M.8.11 Analyze and solve pairs of simultaneous linear equations. - M.8.10 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similarity between them. - M.8.19 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. - M.8.18 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - M.8.17 Verify experimentally the properties of rotations, reflections and translations: - M.8.16 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. - M.8.15 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - M.8.14 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. (e.g., The function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.) - M.8.13 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.) - M.8.12 Use linear functions to represent and explain real-world and mathematical situations. - PA.A.1.2 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - 8.G.A.4 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 8.G.8 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - M.8.22 Recognize that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. - PA.A.1.1 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. - 8.G.A.5 Explain a proof of the Pythagorean Theorem and its converse. - M.8.21 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 8.G.7 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. (e.g., Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.) - M.8.20 Explain a proof of the Pythagorean Theorem and its converse. - 8.G.6 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - 8.G.A.2 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. (e.g., Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.) - M.8.7 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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - M.8.8 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - 8.G.A.3 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. - 8.G.5 Identify a function as linear if it can be expressed in the form y = mx + b or if its graph is a straight line. - PA.A.1.3 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 8.G.9 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. (e.g., Collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?) - M.8.28 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (e.g., In a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.) - M.8.27 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. - M.8.26 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - 8.G.4 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - 8.G.3 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. - M.8.25 Verify experimentally the properties of rotations, reflections, and translations: - 8.G.A.1 Know the formulas for the volumes of cones, cylinders and spheres and use them to solve real-world and mathematical problems. - M.8.24 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - 8.G.2 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - M.8.23 Verify experimentally the properties of rotations, reflections, and translations: - 8.G.1 Recognize that the graph of a linear function has a constant rate of change - 8.F.A.3b Give examples of nonlinear functions. - 8.F.A.3c Interpret the equation y = mx + b as defining a linear function, whose parameters are the slope (m) and the yintercept (b). - 8.F.A.3a Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. - 8.EEI.A.4b Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. - 8.EEI.A.4a Translate among the multiple representations of a function, including mappings, tables, graphs, equations, and verbal descriptions. - 8.F.1c Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. - 8.EE.8a Determine if a relation is a function using multiple representations, including mappings, tables, graphs, equations, and verbal descriptions. - 8.F.1d Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. - 8.EE.8b Compare properties of two functions, each represented in a different way (i.e., algebraically, graphically, numerically in tables, or by verbal descriptions). - M08.B-F.1.1.2 Graph a function from a table of values. Understand that the graph and table both represent a set of ordered pairs of that function. - 8.F.1e Interpret the equation of y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. - M08.B-F.1.1.3 Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. - 8.EE.8c Determine whether a relation is a function. - M08.B-F.1.1.1 Understand that a function assigns to each input exactly one output. - 8.F.1a Relate inputs (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥-values or domain) and outputs (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦-values or range) to independent and dependent variables. - 8.F.1b Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. - MAFS.8.G.1.4 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. - MAFS.8.G.1.3 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. - MAFS.8.G.1.5 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. - MAFS.8.G.1.2 Verify experimentally the properties of rotations, reflections, and translations: - MAFS.8.G.1.1 Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. - 8.1.1.4 Convert a terminating or repeating decimal to a rational number (limit repeating decimals to thousandths). - M08.A-N.1.1.2 Determine whether a number is rational or irrational. For rational numbers, show that the decimal expansion terminates or repeats (limit repeating decimals to thousandths). - M08.A-N.1.1.1 Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. - 8.1.1.5 Use the Pythagorean Theorem to find the distance between points in a Cartesian coordinate system. - 8.GM.B.8 Use the Pythagorean Theorem to determine unknown side lengths in right triangles in problems in two- and three-dimensional contexts. - 8.GM.B.7 Use models to demonstrate a proof of the Pythagorean Theorem and its converse. - 8.GM.B.6 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. - MAFS.8.NS.1.1 Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. - 8.1.1.1 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). For example, by truncating the decimal expansion of the square root of 2, show that the square root of 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - MAFS.8.NS.1.2 Locate/identify rational and irrational numbers at their approximate locations on a number line. - M08.A-N.1.1.5 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. - 8.SP.3 Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. - 8.1.1.2 Use rational approximations of irrational numbers to compare and order irrational numbers. - M08.A-N.1.1.4 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? - 8.SP.4 Determine rational approximations for solutions to problems involving real numbers. - 8.1.1.3 Estimate the value of irrational numbers without a calculator (limit whole number radicand to less than 144). - M08.A-N.1.1.3 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. - 8.SP.1 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. - 8.SP.2 Represent linear functions with tables, verbal descriptions, symbols, and graphs; translate from one representation to another. - PA.A.2.1 Solve problems involving linear functions and interpret results in the original context. - PA.A.2.5 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 8.G.B.7 Predict the effect on the graph of a linear function when the slope or y-intercept changes. Use appropriate tools to examine these effects. - PA.A.2.4 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 8.G.B.8 Identify graphical properties of linear functions including slope and intercepts. Know that the slope equals the rate of change, and that the yintercept is zero when the function represents a proportional relationship. - PA.A.2.3 Identify, describe, and analyze linear relationships between two variables. - PA.A.2.2 Explain a proof of the Pythagorean Theorem and its converse. - 8.G.B.6 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. - 8.F.1 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. - 8.F.5 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - 8.F.4 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. - 8.F.3 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. - 8.F.2 The student will make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs. - 8.16e The student will determine the independent and dependent variable, given a practical situation modeled by a linear function. - 8.16c The student will graph a linear function given the equation in y = mx + b form. - 8.16d Mathematical Modeling: Analyze and Solve Linear Equations - 8.AF.2MM 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. - MAFS.8.F.2.5 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. - MAFS.8.F.2.4 The student will recognize and describe the graph of a linear function with a slope that is positive, negative, or zero. - 8.16a The student will identify the slope and y-intercept of a linear function, given a table of values, a graph, or an equation in y = mx + b form. - 8.16b Find an approximate equation for the line of best fit using two appropriate data points. - 8.DSP.3a Interpret the slope and intercept. - 8.DSP.3b Solve problems using the equation. - 8.DSP.3c Sketch the graph of a function that has been described verbally. - 8.AF.3.6 Define an equation in slope-intercept form (𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦=𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥+𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏) as being a linear function. - 8.F.3a Recognize that the graph of a linear function has a constant rate of change. - 8.F.3b Provide examples of nonlinear functions. - 8.F.3c Tell whether a relation is a function. - 8.AF.3.1 Know and apply the properties of integer exponents to generate equivalent expressions. - 8.EEI.A.1 Describe the behavior of a function and write a description to go with its graph. - 8.AF.3.5 Apply the Pythagorean Theorem to model and solve real-world and mathematical problems in two and three dimensions involving right triangles. - 8.GM.7 Write an equation in the form y = mx + b to describe a linear function. - 8.AF.3.4 Use models to demonstrate a proof of the Pythagorean Theorem and its converse. - 8.GM.6 Compare linear and nonlinear functions. - 8.AF.3.3 Solve real-world and mathematical problems involving volumes of cones, cylinders, and spheres and the surface area of cylinders. - 8.GM.9 Express very large and very small quantities in scientific notation and approximate how many times larger one is than the other. - 8.EEI.A.3 Find the distance between any two points in the coordinate plane using the Pythagorean Theorem. - 8.GM.8 Identify functions by their equations, tables, and graphs. - 8.AF.3.2 Informally justify the Pythagorean Theorem using measurements, diagrams, or dynamic software and use the Pythagorean Theorem to solve problems in two and three dimensions involving right triangles. - PA.GM.1.1 Use the Pythagorean Theorem to find the distance between any two points in a coordinate plane. - PA.GM.1.2 Understand that two-dimensional figures are congruent if a series of rigid transformations can be performed to map the preimage to the image. - 8.GM.A.2 Verify experimentally the congruence properties of rigid transformations. - 8.GM.A.1 The student will determine whether a given relation is a function. - 8.15a Understand that two-dimensional figures are similar if a series of transformations (rotations, reflections, translations and dilations) can be performed to map the pre-image to the image. - 8.GM.A.4 Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. - 8.GM.A.3 Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. - 8.2.2.1 Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦-intercept is zero when the function represents a proportional relationship. - 8.2.2.2 Identify how coefficient changes in the equation f(x) = mx + b affect the graphs of linear functions. Know how to use graphing technology to examine these effects. - 8.2.2.3 Analyze and solve pairs of simultaneous linear equations. - MAFS.8.EE.3.8 Mathematical Modeling: Use Functions to Model Relationships - 8.AF.3MM Verify experimentally parallel lines are mapped to parallel lines. - NY-8.G.1c Verify experimentally angles are mapped to angles of the same measure. - NY-8.G.1b Identify congruent and supplementary pairs of angles when two parallel lines are cut by a transversal. - 8.GM.5c Recognize that two similar figures have congruent corresponding angles. - 8.GM.5d Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. - NC.8.SP.1 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. - MAFS.8.F.1.3 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. - MAFS.8.F.1.2 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. - MAFS.8.F.1.1 Analyze and describe attributes of graphs of functions (e.g., constant, increasing/decreasing, linear/nonlinear, maximum/minimum, discrete/continuous). - 8.F.5a Collect bivariate data. - 8.DSP.1a Sketch the graph of a function from a verbal description. - 8.F.5b Use the equation of a linear model to solve problems in the context of bivariate quantitative data, interpreting the slope and y-intercept. - NC.8.SP.3 Write a verbal description from the graph of a function with and without scales. - 8.F.5c Graph the bivariate data on a scatter plot. - 8.DSP.1b Describe patterns observed on a scatter plot, including clustering, outliers, and association (positive, negative, no correlation, linear, nonlinear). - 8.DSP.1c Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. - NC.8.SP.4 Verify experimentally lines are mapped to lines, and line segments to line segments of the same length. - NY-8.G.1a Calculate the surface area of a cylinder, in terms of π and using approximations for π, using decomposition or nets. Use appropriate measurements such as cm². - PA.GM.2.2 Develop and use the formulas V = πr2ℎ and V = Bℎ to determine the volume of right cylinders, in terms of π and using approximations for π. Justify why base area (B) and height (h) are multiplied to find the volume of a right cylinder. Use appropriate measurements such as cm³. - PA.GM.2.4 Recognize that two-dimensional figures are only similar if a series of transformations can be performed to map the pre-image to the image. - 8.GM.4b Given two similar figures, describe the series of transformations that justifies this similarity. - 8.GM.4c Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as f(x), to represent such relationships. - 8.2.1.1 Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. - 8.2.1.2 Understand that the slope is the constant rate of change and the 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦-intercept is the point where 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦𝑥 = 0. - 8.F.4a Determine the slope and the 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦𝑥𝑦-intercept of a linear function given multiple representations, including two points, tables, graphs, equations, and verbal descriptions. - 8.F.4b Construct a function in slope-intercept form that models a linear relationship between two quantities. - 8.F.4c Lines are taken to lines, and line segments to line segments of the same length. - 8.G.1a Understand that a function is linear if it can be expressed in the form f(x) = mx + b or if its graph is a straight line. - 8.2.1.3 Interpret the meaning of the slope and the 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦𝑥𝑦𝑦-intercept of a linear function in the context of the situation. - 8.F.4d Angles are taken to angles of the same measure. - 8.G.1b Discover that the sum of the three angles in a triangle is 180 degrees. - 8.GM.5a Parallel lines are taken to parallel lines. - 8.G.1c Discover and use the relationship between interior and exterior angles of a triangle. - 8.GM.5b Use the Pythagorean Theorem to solve problems involving right triangles. - 8.3.1.1 Determine the distance between two points on a horizontal or vertical line in a coordinate system. Use the Pythagorean Theorem to find the distance between any two points in a coordinate system. - 8.3.1.2 Informally justify the Pythagorean Theorem by using measurements, diagrams and computer software. - 8.3.1.3 The student will solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable. - 8.17 Use coordinate geometry to describe the effect of transformations on two-dimensional figures. - 8.GM.3a Relate scale drawings to dilations of geometric figures. - 8.GM.3b Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. - MAFS.8.EE.3.8a Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. - MAFS.8.EE.3.8b Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. - MAFS.8.EE.3.8c Use scientific notation to write very large or very small quantities. - 8.AF.1.9 Estimate large and small quantities using a power of 10. - 8.AF.1.8 Dilate geometric figures using scale factors that are positive rational numbers. - 8.GM.4a Compare and order rational and irrational numbers. - 8.AF.1.3 Identify a number that is irrational. - 8.AF.1.2 Write repeating decimals as fractions. - 8.AF.1.1 Write a number with a negative or zero exponent a different way. - 8.AF.1.7 Use the properties of exponents to write equivalent expressions. - 8.AF.1.6 Solve equations involving squares or cubes. - 8.AF.1.5 Find square roots and cube roots of rational numbers. - 8.AF.1.4 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² ×3-5=1/3³=1/27 - MAFS.8.EE.1.1 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 10⁸ and the population of the world as 7 × 10⁹ , and determine that the world population is more than 20 times larger. - MAFS.8.EE.1.3 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - MAFS.8.EE.1.2 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. - MAFS.8.EE.1.4 Rotate geometric figures 90, 180, and 270 degrees, both clockwise and counterclockwise, about the origin. - 8.GM.2a Reflect geometric figures with respect to the 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦𝑥𝑦𝑦𝑥-axis and/or 𝑦𝑥𝑝𝑝𝑝𝑎𝑝𝑎𝑏𝑦𝑘𝑥𝑦𝑚𝑥𝑏𝑏𝑦𝑚𝑥𝑏𝑚𝑥𝑎𝑎𝑎𝑎𝑏𝑥𝑦𝑦𝑚𝑥𝑏𝑦𝑦𝑥𝑦𝑦𝑥𝑦-axis. - 8.GM.2b Translate geometric figures vertically and/or horizontally. - 8.GM.2c Recognize that two-dimensional figures are only congruent if a series of rigid transformations can be performed to map the pre-image to the image. - 8.GM.2d Given two congruent figures, describe the series of rigid transformations that justifies this congruence. - 8.GM.2e Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. - 8.2.4.2 Make sense of problems and persevere in solving them. 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. - MAFS.K12.MP.1.1 Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. - 8.2.4.3 Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships. - 8.2.4.1 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 = a, a = a, or a = b results (where a and b are different numbers). - M.8.9a Represent relationships in various contexts using systems of linear equations. Solve systems of linear equations in two variables symbolically, graphically and numerically. - 8.2.4.7 Understand that a system of linear equations may have no solution, one solution, or an infinite number of solutions. Relate the number of solutions to pairs of lines that are intersecting, parallel or identical. Check whether a pair of numbers satisfies a system of two linear equations in two unknowns by substituting the numbers into both equations. - 8.2.4.8 Use the relationship between square roots and squares of a number to solve problems. - 8.2.4.9 Understand 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. - NY-8.SP.2 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. - NY-8.SP.1 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. - NY-8.SP.3 Mathematical Modeling: Real Numbers - 8.AF.1MM Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - M.8.9b Verify that lines are mapped to lines, including parallel lines. - 8.GM.1a Verify that corresponding angles are congruent. - 8.GM.1b Verify that corresponding line segments are congruent. - 8.GM.1c Derive the equation y = mx + b. - 8.AF.2.9 Find the y-intercept of a graph and explain what it means. - 8.AF.2.8 Write equations to describe linear relationships. - 8.AF.2.7 Solve equations with variables on both sides of the equal sign. - 8.AF.2.2 Solve equations that have like terms on one side. - 8.AF.2.1 Understand the slope of a line. - 8.AF.2.6 Compare proportional relationships represented in different ways. - 8.AF.2.5 Determine the number of solutions an equation has. - 8.AF.2.4 Solve multistep equations and pairs of equations using more than one approach. - 8.AF.2.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. - MAFS.8.SP.1.3 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? - MAFS.8.SP.1.4 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 y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - MAFS.8.EE.2.6 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - MAFS.8.EE.2.5 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. - MAFS.8.SP.1.1 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. - MAFS.8.SP.1.2 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 = a, a = a, or a = b results (where a and b are different numbers). - MAFS.8.EE.3.7a Organize bivariate categorical data in a two-way table. - 8.DSP.4a Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - MAFS.8.EE.3.7b Interpret data in two-way tables using relative frequencies. - 8.DSP.4b Explore patterns of possible association between the two categorical variables. - 8.DSP.4c The student will compare and order real numbers. - 8.1 The student will describe the relationships between the subsets of the real number system. - 8.2 List of all Files Validated: imsmanifest.xml I_0003d843-c6b0-317f-85d7-4640667398f7_R/BasicLTI.xml I_006a518d-5c73-3b04-93fe-87a8811190b6_1_R/BasicLTI.xml I_006a518d-5c73-3b04-93fe-87a8811190b6_R/BasicLTI.xml I_0070aea8-599d-3e98-b50a-3de337d6360d_1_R/BasicLTI.xml I_007e98b2-7c3b-33e9-9742-0eb7f334c956_R/BasicLTI.xml I_0081830a-420d-3ab9-a683-39d12ad8c3dd_R/BasicLTI.xml I_00819f6d-9722-392c-94d6-9c7babca0735_1_R/BasicLTI.xml I_00840f78-cb51-3d1d-a7cc-704f58d38070_1_R/BasicLTI.xml I_0090f68d-3a28-35c8-8ce4-8773cd4aa09d_1_R/BasicLTI.xml I_00b3a397-f055-3ab4-a1f1-0410ff1fc6ee_1_R/BasicLTI.xml I_00cea13f-e592-3f39-a31f-cb41e84fffa1_R/BasicLTI.xml I_00e7be13-8708-32a3-b8b1-9fee620c0681_R/BasicLTI.xml I_00ece8f4-a87a-345b-ab69-d4d57e5417d7_1_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_1_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_2_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_3_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_4_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_5_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_6_R/BasicLTI.xml I_00f767a3-6627-3f51-93b0-cae05b5f68ae_R/BasicLTI.xml I_010b0426-3326-3acf-a0b7-14e85736bbb5_1_R/BasicLTI.xml I_0135e5e8-0729-3a38-a301-45ed4785e16c_R/BasicLTI.xml I_0137c331-d85f-3c13-a3ac-09dbc643ffe6_1_R/BasicLTI.xml I_017563a4-6e2b-38a7-ae8b-9f190e192c35_1_R/BasicLTI.xml I_017ca5cd-83f8-3115-8936-7b7b6778aca0_1_R/BasicLTI.xml I_019173b6-fac7-3a0d-8613-49dcf19239d8_1_R/BasicLTI.xml I_01b9d7f6-0a5d-3993-9e37-eeed789cf8b4_1_R/BasicLTI.xml I_01ca8d81-1cf8-39fd-a430-5db39eb1e354_R/BasicLTI.xml I_01cd3d9d-11f5-3970-91b9-29d6bdfabeec_R/BasicLTI.xml I_01cefdc3-4222-308c-b18e-a461a0fa923b_1_R/BasicLTI.xml I_01e57c85-511a-3bc5-aebb-3230f1b368ac_1_R/BasicLTI.xml I_01ea06a4-c8b8-3662-a4ac-178f391742dc_R/BasicLTI.xml I_01eca747-8875-312e-973e-4f2b53dcc532_R/BasicLTI.xml I_01fce3a8-b53d-31ef-a3ff-043f543f5d29_1_R/BasicLTI.xml I_02143fa3-1572-3b4d-8cfc-9ae3c82532ad_1_R/BasicLTI.xml I_02143fa3-1572-3b4d-8cfc-9ae3c82532ad_3_R/BasicLTI.xml I_021d44ac-4024-352b-b07c-625c2167ea7b_1_R/BasicLTI.xml I_022a9a1f-9ef7-3942-b072-99d01ec47003_R/BasicLTI.xml I_0240fdd5-7ad8-3a80-b89c-57fbdfb5adac_1_R/BasicLTI.xml I_0246cdfb-3d86-3071-b247-b67c35454c7b_R/BasicLTI.xml I_025037c3-2813-309b-9c41-6f774c50c550_1_R/BasicLTI.xml I_0253bc74-7d42-381a-b911-b3b20a69b6a2_1_R/BasicLTI.xml I_02675a80-8643-3b01-b70e-746651194f0b_1_R/BasicLTI.xml I_02686f57-955f-349e-b69c-5cef074515a3_1_R/BasicLTI.xml I_0281aed2-ca76-3e31-a63c-567a1a683daa_1_R/BasicLTI.xml I_028ed569-7201-3248-aa44-6f8ecf9c4847_1_R/BasicLTI.xml I_02aa7c15-5ffb-355e-bf6a-7fae99b67143_1_R/BasicLTI.xml I_02aecd66-47c9-3ae9-97bd-15037433d608_R/BasicLTI.xml I_02c7d455-6db0-3944-b99d-fb861f441489_1_R/BasicLTI.xml I_02cda9ae-eca8-3672-bcc9-50766a1952c4_1_R/BasicLTI.xml I_03047ac6-09da-39de-a07a-d20f2fcddc92_1_R/BasicLTI.xml I_03077d60-ba05-32f9-9eda-a2045443c7b3_1_R/BasicLTI.xml I_03286207-88c1-3fc3-9dd9-9a2e8c5bacd8_1_R/BasicLTI.xml I_0330a903-5125-3916-a2ef-d0e83664b103_R/BasicLTI.xml I_03384eb3-3798-3c4b-9a35-53b5d828f8c9_1_R/BasicLTI.xml I_0338c842-88af-35b6-bef4-2bb44219d05f_R/BasicLTI.xml I_0341f7a7-0d34-396b-a882-c66ea109c3cf_R/BasicLTI.xml I_034e4a84-fb09-3b2c-85ce-00afa78b95d1_1_R/BasicLTI.xml I_036a02e9-3166-34a7-854a-0bf274b41ac7_1_R/BasicLTI.xml I_03722541-bff6-3a38-bfde-66cd976e8b31_1_R/BasicLTI.xml I_03950398-9b80-349c-ac7f-c5762809fd61_1_R/BasicLTI.xml I_039691c5-9537-376f-a5a9-26c875848a0f_1_R/BasicLTI.xml I_039691c5-9537-376f-a5a9-26c875848a0f_3_R/BasicLTI.xml I_03973662-044e-3498-8562-2f874a8b43f6_1_R/BasicLTI.xml I_03a2e48f-61cd-3517-9df2-73657cbcae0b_R/BasicLTI.xml I_03c318ed-b166-33a3-ab78-5c88f664367c_1_R/BasicLTI.xml I_03c88bb1-536d-3183-8e21-bbbd5b337411_1_R/BasicLTI.xml I_03da0a41-3277-3b1d-92d8-ee6b35992d9f_1_R/BasicLTI.xml I_03de9013-f1aa-36e3-9469-329828655996_1_R/BasicLTI.xml I_03f3e44d-61f0-3129-83b6-70a18da96a03_1_R/BasicLTI.xml I_04055969-4282-383a-9b84-01b31e787183_R/BasicLTI.xml I_04192c6c-144a-3c82-a1c3-96a0ff27b5c8_1_R/BasicLTI.xml I_043916d7-d7e0-3166-b38e-359e11c6822f_R/BasicLTI.xml I_043b8bca-7f71-37d4-8566-4c01036eb6a0_1_R/