I can demonstrate for rational numbers that the decimal expansion repeats eventually.

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1 Grade 8 Math Length of Class: School Year Program/Text Used: Competency 1: The Number Systems - Students will demonstrate the ability to know that there are numbers that are not rational and approximate them by rational numbers. Performance Advanced Indicator(s) (In addition to B ) Students will know that numbers that are not rational are called irrational, understand informally that every number has a decimal expansion, and, for rational numbers, show that the decimal expansion repeats eventually. I can convert a decimal expansion which repeats eventually into a rational number. I can demonstrate for rational numbers that the decimal expansion repeats eventually. I can understand that numbers that are not rational are called irrational. I can recognize that 2 is irrational. I can understand informally that every number has a decimal expansion. I inconsistently know that numbers that are not rational are called irrational. I can understand informally that every number has a decimal expansion Students will be able to calculate rational approximations of irrational numbers to compare the size of irrational numbers. I can explain how to continue on to get better approximations. I can estimate the value of expressions (e.g., 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5. I can use rational approximations of irrational numbers to compare the size of irrational numbers and locate them approximately on a number line diagram. I can use rational approximations or irrational numbers to compare the size of irrational numbers and locate them on a number line diagram

2 Competency 2: Expressions and Equations - Students will demonstrate the ability to work with radicals and integer exponents; understand the connections between proportional relationships, lines, and linear equations; and analyze and solve linear equations and pairs of simultaneous linear equations. Performance Indicator(s) Advanced Students will know and apply the properties of integer exponents to generate equivalent numerical expressions. Students will use square 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. Students will be able to express numbers using scientific notation. I can prove the properties of integer exponents. I can express how many times as much one is than the other. 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. I can apply the properties of integer exponents to generate equivalent numerical expressions. I can 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. I can perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. I can interpret scientific notation that has been generated by technology. I can recognize the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/3 3 = 1/27. I can evaluate square roots of small perfect squares and cube roots of small perfect cubes. I can use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities. I can recognize the properties of integer exponents to generate equivalent numerical expressions I can evaluate square roots of small perfect squares and cube roots of small perfect cubes I can use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities

3 Competency 2: Expressions and Equations, cont. Performance Indicator(s) Advanced Students will graph I can compare two proportional relationships. 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 Students will 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. Students will recognize the slope and y-intercept of a line given its equation y = mx + b. Students can create the equation y = mx + b given the graph of a line. speed. I can 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. I can derive the equation of y = mx + b given the coordinates of two points on the line or the two intercepts or any combination thereof. I can interpret the unit rate as the slope of the graph. I can 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. I can create the equation of y = mx + b given the coordinates of a point and the line s slope. I can create the equation of a line from a graph. I can graph proportional relationships. I can recognize that similar triangles can be used to show the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane. I can recognize the slope and y-intercept of a line given its equation y = mx + b. I can create the equation y = mx + b given the graph of a line. I can graph proportional relationships I can recognize that similar triangles can be used to show the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane inconsistently or I can recognize the slope and y-intercept of a line given its equation y = mx + b incompletely or

4 Competency 2: Expressions and Equations, cont Performance Indicator(s) Advanced Students will identify I can apply the various linear equations in one solution types to variable with one solution, appropriate real-life infinitely many solutions, situations. or no solutions and show which of these possibilities is the case. Students will solve linear equations in one variable with integer coefficients, including equations whose solutions require either expanding expressions using the distributive property or combining like terms with the variable on one side. I can solve linear equations in one variable with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms with the variable on both sides of the equation. I can solve linear equations in one variable with one solution, infinitely many solutions, or no solutions and explain 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). I can solve linear equations in one variable with integer coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms with the variable on both sides of the equation. I can identify linear equations in one variable with one solution, infinitely many solutions, or no solutions and show which of these possibilities is the case. I can solve linear equations in one variable with integer coefficients, including equations whose solutions require either expanding expressions using the distributive property or combining like terms with the variable on one side. I can identify linear equations in one variable with one solution, infinitely many solutions, or no solutions and show which of these possibilities is the case I can solve linear equations in one variable with integer coefficients, including equations whose solutions require either expanding expressions using the distributive property or combining like terms with the variable on one side

5 Competency 2: Expressions and Equations cont Performance Indicator(s) Advanced Students will understand I can solve real-world and that solutions to a system mathematical problems of two linear equations in leading to two linear two variables correspond equations in two variables. to points of intersection of For example, given their graphs, because coordinates for two pairs points of intersection of points, I can determine satisfy both equations whether the line through simultaneously. the first pair of points intersects the line through the second pair using any method of solving systems of linear equations. I can solve systems of two linear equations in two variables algebraically and explain what the solution means. I can identify the solution type: one solution, infinitely many solutions, or no solution, given a graph of a pair of simultaneous linear equations. I can estimate solutions by graphing the equations. I can 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. I can identify the solution type: one solution, infinitely many solutions, or no solution, given a graph of a pair of simultaneous linear equations, inconsistently or I can inaccurately estimate solutions by graphing the equations. I can solve simple cases by inspection inconsistently or

6 Competency 3: Functions - Students will demonstrate the ability to define, evaluate, and compare functions as well as use functions to model relationships between quantities. Performance Indicator(s) Students will define that a function is a rule that assigns to each input exactly one output; understand the graph of a function is the set of ordered pairs consisting of an input and the corresponding output; and interpret the equation y=mx+b as defining a linear function, whose graph is a straight line. Students will 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. Students will describe qualitatively the functional relationship between two quantities by analyzing a graph. Advanced I can analyze examples of functions that are nonlinear. 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. I can construct a function to model a linear relationship between two quantities. I can design real-world scenarios (linear and nonlinear), represent the scenarios graphically and analyze the graphs. I can 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, I can determine which function has the greater rate of change. I can 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. I can construct a graph that exhibits the qualitative features of a linear and nonlinear function that has been described verbally. I can define 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. I can interpret the equation y=mx+b as defining a linear function, whose graph is a straight line. Given various graphs, I can identify which are or are not linear functions. I can 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. I can describe qualitatively the functional relationship between two quantities by analyzing a graph For example, I can identify where the function is increasing or decreasing, linear or nonlinear. I can define that a function is a rule that assigns to each input exactly one output inconsistently or I can interpret the equation y=mx+b as defining a linear function inconsistently or I can determine the rate of change and initial value of the function from a description of a relationship I can describe qualitatively the functional relationship between two quantities by analyzing a graph

7 Competency 4: Geometry - Students will demonstrate the ability to understand congruence, similarity, and angle relationships using physical models, transparencies, or geometry software; understand and apply the Pythagorean Theorem; and solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. Performance Indicator(s) Students will show that a two-dimensional figure is congruent to another through a sequence of rotations, reflections, and translations and describe the sequence that exhibits the congruence between them. Students will 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. Advanced I can identify and analyze the concept of congruency as it relates to a real-world situation. I can identify and analyze the concept of similarity as it relates to a real-world situation. I can create a congruent figure when given the original two-dimensional shape. I can describe the effect of dilations on twodimensional figures using coordinates. I can show that a twodimensional figure is congruent to another through a sequence of rotations, reflections, and translations. Given two congruent figures, I can describe a sequence that exhibits the congruence between them. I can 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 twodimensional figures, I can describe a sequence that exhibits the similarity between them. I can show that a twodimensional figure is congruent to another through a sequence of rotations, reflections, and translations I can 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 inconsistently or

8 Competency 4: Geometry, cont. Performance Indicator(s) Advanced Students will verify experimentally the properties of rotations, reflections, and translations. Students will 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. I can analyze the effect of translations, rotations, and reflections in real-world situations. I can identify and analyze a real-world situation using angle relationships of triangles and parallel lines cut by a transversal. I can describe the effect of translations, rotations, and reflections on twodimensional figures using coordinates. I can determine numerical measures of angles, given algebraic expressions for angle measurements. I can verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. I can 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, I can 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. I can verify experimentally the properties of rotations, reflections, and translations I can 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

9 Competency 4: Geometry, cont. Performance Indicator(s) Students will understand and apply the Pythagorean Theorem. Students will recognize and match formulas for the volumes with the appropriate threedimensional objects of cones, cylinders, and spheres. Advanced (In addition to ) I can explain a formal proof of the Pythagorean Theorem and its converse. I can apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in three dimensions. I can determine the unknown dimension of a cone, cylinder, and sphere, given the volume in a realworld situation. I can apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions. I can apply the Pythagorean Theorem to find the distance between two points in a coordinate system. I can use the formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems. I can state the Pythagorean Theorem and its converse using words and symbols. I can determine if a triangle is a right triangle. I can explain an informal proof of the Pythagorean Theorem and its converse. I can recognize and match formulas for the volumes with the appropriate threedimensional objects of cones, cylinders, and spheres. I can state the Pythagorean Theorem and its converse using words and symbols with error. I can determine if a triangle is a right triangle, or explain an informal proof of the Pythagorean Theorem and its converse, I can recognize and match formulas for the volumes with the appropriate threedimensional objects of cones, cylinders, and spheres

10 Competency 5: Statistics and Probability - Students will demonstrate the ability to investigate patterns of association in bivariate data. Performance Indicator(s) Students will interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Students will understand that straight lines are widely used to model relationships between two quantitative variables. Advanced I can collect bivariate measurement data, construct, and interpret scatter plots to investigate patterns of association between two quantities. I can formally fit a straight line and formally assess the model fit for scatter plots that suggest a linear association. I can construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. I can informally fit a straight line and informally assess the model fit by judging the closeness of the data point to the line for scatter plots that suggest a linear association. I can interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. I can describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. I can understand that straight lines are widely used to model relationships between two quantitative variables. I can 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, I can 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 a mature plant height. I can interpret and describe scatter plots for bivariate measurement data to investigate patterns of association between two quantities I can understand that straight lines are widely used to model relationships between two quantitative variables I can use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept

11 Competency 5: Statistics and Probability, cont. Performance Indicator(s) Advanced (In addition to ) Students will understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table and use relative frequencies to describe possible association between the two variables. I can calculate relative frequencies for rows and columns to describe possible association between the two variables. I can construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. For example, I can collect data from students in [your] my 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? I can understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. I can use relative frequencies calculated for rows or columns to describe possible association between the two variables. I can understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table I can use relative frequencies calculated for rows or columns to describe possible association between the two variables Competency 6: Fact Fluency - Students will demonstrate the ability to quickly and accurately verbalize and compute fact fluency. Performance Indicator(s) Students will be able to demonstrate the ability to accurately and efficiently perform basic mathematical skills. I can demonstrate fluency with addition, subtraction, multiplication, and division facts both mentally and with paper and pencil. I inconsistently demonstrate fluency with addition, subtraction, multiplication, or division facts both mentally and with paper and pencil.

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