# Mathematics Grade-Level Instructional Materials Evaluation Tool

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4 I. CCSS-M CLUSTERS AND STANDARDS GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.1-4. Materials demonstrate how to work with radicals and integer exponents by showing and explaining how to: Apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3-3 = = uurequiring the rule 10 a 10 b = 10 a+b to hold when a and b are integers leads to the definition of the meaning of powers with 0 and negative exponents. For example, we define 10 0 = 1 because we want 10 a x 10 0 = 10 a+0 = 10 a, so 10 0 must equal 1. Students extend these rules to other bases, and learn other properties of exponents. This lays the foundation for students to learn the properties of rational exponents in high school. Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p and explain why 2 is irrational. uusince p is defined as the positive solution to the equation x 2 =p (when it exists), it is not correct to say (as is common) that 64 = ±8. On the other hand, in describing the solutions to x 2 = 64, students can write x = ± 64 = ±8. Use numbers expressed in the form of a single digit times a whole-number 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, they estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. Perform operations with, estimate, and interpret numbers expressed in scientific notation, using appropriate size for measurements of very large or very small quantities. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 4

5 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.1-4. Assignments ask students to work with radicals and integer exponents by: Evaluating square roots of small perfect squares and cube roots of small perfect cubes. Knowing and describing why 2 is irrational. Evaluating and generating equivalent numerical expressions by applying properties of integer exponents. Solving equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Estimating quantities using scientific notation and determining how many times as large one number is in relation to another. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. Performing operations with, estimating, and interpreting numbers expressed in scientific notation, using appropriate size for measurements of very large or very small quantities. uufor example, given that we breathe about 6 liters of air per minute, they estimate that there are = minutes in a day, and that we therefore breathe about liters in a day. In a lifetime of 75 years there are about days, and so we breathe about = liters of air in a lifetime. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 5

6 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.5-6. Materials illustrate connections between proportional relationships, lines, and linear equations by showing and explaining how to: Graph proportional relationships, interpret the unit rate as the slope of the graph of a proportional relationship, and apply these concepts to solve realworld problems. Compare two different proportional relationships represented in different ways. uuin grade eight, proportional relationships should prompt students to begin connecting a number of notions that have been developing over grades six and seven: 1) an expression in one variable defines a general calculation in which the variable can represent a range of numbers with the variable representing the input and the expression calculating the output; 2) choosing a variable to represent the output leads to an equation in two variables describing the relationship between two quantities; 3) tabulating values of the expression is the same as tabulating solution pairs of the corresponding equation; and 4) plotting points on the coordinate plane affords a visual representation of the relationship between two quantities. For example, a table shows 300 miles in 5 hours, whereas the graph could show more than 300 miles in the same time. Or, one can compare a distance-time graph to a distance-time equation to determine which of the two moving objects has greater speed. Use similar triangles to show that the slope is the same between any two distinct points on a non-vertical line in the coordinate plane. Show that whether a line has a well-defined slope that the ratio between the rise and run for any two points on the line is always the same depends on similar triangles. uuthe connection between the unit rate in a proportional relationship and the slope of its graph depends on a connection with the geometry of similar triangles. 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. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 6

7 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.5-6. Assignments ask students to illustrate connections between proportional relationships, lines, and linear equations by: Graphing proportional relationships, interpreting the unit rate as the slope of the graph of a proportional relationship, and applying these concepts to solve real-world problems. Comparing 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. Using similar triangles to show that the slope 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. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 7

8 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.7-8. Materials demonstrate how to analyze and solve linear equations and pairs of simultaneous linear equations by showing and explaining how to: Transform equations into simple forms, providing 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). Solve mathematical and real-world linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining like terms. Analyze and solve pairs of simultaneous linear equations. 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. Solve systems of two linear equations in two variables algebraically, estimate solutions by graphing the equations, and 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. Solve mathematical and real-world 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. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 8

9 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.EE.7-8. Assignments ask students to analyze and solve linear equations by: Solving mathematical and real-world linear equations in one variable, with rational number coefficients, including those that require use of the distributive property or combining of like terms. Analyzing and solving mathematical and real-world problems leading to two linear equations in two variables graphically, algebraically, and by inspection. uulinear equations arise when two linear functions are compared. For example, Henry and Jose are gaining weight for football. Henry weighs 205 pounds and is gaining 2 pounds per week. Jose weighs 195 pounds and is gaining 3 pounds per week. When will they weigh the same? uusolve problems leading to simultaneous equations. For example, tickets for the class show are \$3 for students and \$10 for adults. The auditorium holds 450 people. The show was sold out and the class raised \$2750 in ticket sales. How many students bought tickets? 8.F.1-3. Materials demonstrate how to define, evaluate, and compare functions by: Showing and explaining 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 (function notation is not required in grade eight). Explaining and showing how to compare properties of two functions in multiple ways (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. Showing and explaining how to interpret the equation y = mx + b as a linear function, whose graph is a straight line. Providing 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. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 9

10 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.F.1-3. Assignments ask students to define, evaluate, and compare functions by: Determining whether a graph, table, or algebraic representation is a function. This extends prior experiences with graphs and tables as students explore functional relationships algebraically, graphically, numerically in tables, and through verbal descriptions. They explain correspondences between verbal descriptions, tables, and graphs. Repeated reasoning about entries in tables or points on graphs results in equations for functional relationships. Classifying, comparing, and describing properties of two functions represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Interpreting the equation y = mx + b as defining a linear function, whose graph is a straight line; giving examples of functions that are not linear. 8.F.4-5. Materials demonstrate how to use functions to model relationships between quantities by showing and explaining how to: Use a function to model a linear relationship between two quantities. Determine the rate of change and initial value of a 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. Identify 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. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 10

11 GUIDING STATEMENTS SPECIFIC EVIDENCE FROM THE TEXT/MATERIALS 8.F.4-5. Assignments ask students to use functions to model relationships between quantities by: Constructing a function to model a linear relationship between two quantities described with or without a context. uuwhen using functions to model a linear relationship between two quantities, students need to learn to determine the rate of change of the function, which is the slope of the line that is its graph. They can read (or compute or approximate) the rate of change from a table or a graph. Determining and interpreting 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. Describing qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketching a graph that exhibits the qualitative features of a function that has been described verbally. uuto understand relationships between quantities, it is often helpful to describe the relationships qualitatively, paying attention to the general shape of the graph without concern for specific numerical values. The standard approach proceeds from left to right, describing what happens to the output as the input value increases. MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 11

18 NOT RECOMMENDED PROGRAM NAME/EDITION: COMMENTS: Completed by: Date: MATHEMATICS GRADE-LEVEL INSTRUCTIONAL MATERIALS EVALUATION TOOL: QUALITY REVIEW GRADE 8 18

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