Highland Park Public School District

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1 435 Mansfield Street Highland Park, New Jersey Curriculum Guide Mathematics 8 Version 1 Slated for Revision Final Review 6/30/2017

2 Mastery Skills Students will be able to understand, explain, and apply the following concepts Skill Use mathematical vocabulary from previous grades. and skills upon completion of this course: Standard (if applicable) Standards from previous grades, prerequisite to 8 th grade work. Add, subtract, multiply, and divide numbers, which are integers, fractions, percent s, and decimals which may be expressed as fractions (rational numbers), including negative numbers and zero. Locate rational numbers on the number line. Locate ordered pairs of numbers which are integers including negative numbers and zero, fractions, percent s, and decimals (rational numbers), on the coordinate plane. Write the decimal expansion of any fraction by using long division with the standard algorithm, and understand that the decimal expansion found by long division will repeat or end. Write any terminating or repeating decimal as a fraction, using their knowledge of place values for terminating decimals, or an algebraic algorithm for repeating decimals. Know that numbers that are not rational are called irrational. Know that irrational numbers are non-terminating, non-repeating decimals with no fraction representation. Know that pi and most square and cube roots are irrational. Use decimal approximations of irrational numbers to compare them to each other, as decimals on the number line. Use decimal approximations of irrational numbers to operate with them. 8.NS.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.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi squared). For example, by truncating the decimal expansion of, show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 8.EE.2 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 is irrational. 8.EE.3 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. 8.EE.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. Use square root and cube root symbols to

3 represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number, especially those equations generated by the Pythagorean Theorem. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Use numbers expressed in the form of a single digit times an integer power of 10 (scientific notation) to represent very large or very small quantities. Convert numbers of any form to scientific notation. Apply scientific notation to information about Global Issues or Science to enable comparisons. 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. Express how many times as much one number in scientific notation is than another. Express how much larger or smaller (as a difference) one number in scientific notation is than another. 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 Interpret scientific notation that has been generated by technology. Use zero and negative exponents in scientific notation before abstract study of exponent properties has begun. Know and apply the properties of integer exponents to generate equivalent numerical expressions, generalized from properties noted during the study of scientific notation. Use zero and negative exponents for any base, including bases, which are variables. Simplify expressions, which use exponents, 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

4 especially those in scientific notation, but also for numerical expressions in other bases and abstract algebraic expressions. Understand and apply the Order of Operations when simplifying numerical or algebraic expressions Understand that a function is a rule that assigns to each input exactly one output. Understand that a function is a relation, and that most learning comes from observing relationships. Understand that a relation is a set of ordered pairs. Understand that there are non-mathematical functions, which are useful to consider. Understand that functions may be represented as tables, graphs, equations, sets of ordered pairs, verbal rules, and visual descriptions known as mappings. Understand that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Understand that relations may be represented as tables, graphs, equations, sets of ordered pairs, verbal rules, visual descriptions known as mappings, and understand the quality (one-to-one correspondence), which makes a relation a function. Switch among the various representations of functions, especially for comparing their properties. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Represent proportional relationships as tables, graphs, equations. Graph proportional relationships, interpreting the slope of the graph as the unit rate. 8.F.3 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. 8.F.4 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.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation 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.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.F.3 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. 8.F.4 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.

5 Compare two different proportional relationships represented in different ways. Describe the graph of a non-vertical line through the origin using the equation y = mx or y = mx + 0. Describe the table of a horizontal or vertical line. Create a table of ordered pairs (x, y) for the graph of a line. Create a table of ordered pairs (x, y) for an equation in two variables. Create a graph for a linear equation in two variables. Create a graph for a table of ordered pairs. Distinguish tables of linear functions from tables of functions, which are not linear. Describe the graph of a non-vertical line intercepting the y-axis at b using the equation y = mx + b. Describe the graph of a horizontal or vertical line using the equations y = k or x = k. 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. 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept of the linear model in the context of the given data. Understand slope as rate of change, constant of proportionality, unit rate. Understand, use, and apply the slope formula. 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. Construct a function to model a linear relationship between two quantities. Determine the rate of change (slope) and initial

6 value of the function (y-intercept) 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 (slope) and initial value of the function (y-intercept) of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Use the vocabulary of mathematical functions: increasing, decreasing, linear, nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 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. Know that straight lines are widely used to model relationships between two quantitative variables. Informally fit a straight line (trend line, or line of best fit) to scatter plots that suggest a linear association. 8.EE.7 Solve linear equations in one variable. 8.EE.7 a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x=a, a=a, or a=b results (where a and b are different numbers). 8.EE.7 b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.EE.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.8a 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.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Assess the linear model s fit by judging the closeness of the data points to the line. Use the equation of a linear model to make predictions about bivariate measurements. Use the equation of a linear model to solve problems, interpreting the slope and y-intercept appropriately in the context of bivariate measurement data. Solve linear equations with rational number coefficients in one variable by using inverse transformations. Understand that linear equations may have one, 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: 8.G.1a Lines are taken to lines, and line segments to line segments of the same length. 8.G.1b Angles are taken to angles of the same measure. 8.G.1c Parallel lines are taken to parallel lines. 8.G.2 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.

7 zero, or infinitely many solutions, and recognize which sequence of transformations will lead to which result. Solve linear equations, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve problems with linear equations in one variable. Analyze and solve pairs of simultaneous linear equations. Graph a system of two linear equations on the same coordinate plane, and understand that solution to a system of two linear equations in two variables corresponds to points of intersection of their graphs. Estimate the solution to a system of two linear equations by graphing. Understand that the graphed solution is only an estimate without testing it, and understand the need for algebraic methods to get exact solutions for systems. Solve simple systems of two linear equations in two variables by inspection. Solve a system of two linear equations in two variables by substitution. Solve a system of two linear equations in two variables by elimination, or linear combination. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.4 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.5 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. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.7 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.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems. Solve real world and mathematical problems leading to two linear equations in two variables. Understand that a system of linear equations in two variables may have one, zero, or infinitely many solutions, and understand how this is comparable to the number of solutions for a linear equation of one variable. Verify experimentally the properties of rotations, reflections, and translations. Use the properties of rigid transformations: lines are taken to lines, parallel lines are taken to

8 parallel lines, line segments to line segments of the same length, angles are taken to angles of the same measure. 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 of rigid transformations that exhibits the congruence between them. 8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Is there evidence that those who have a curfew also tend to have chores? Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 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. Use arguments following from properties of transformations 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. Use the vocabulary of geometry for transformations and angle relations. Understand a proof of the Pythagorean Theorem, which uses sums of square areas. Use the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two dimensions. Understand how the Pythagorean Theorem can be used to solve problems by adding imaginary lines to make right triangles on drawings of problems. Apply the Pythagorean Theorem to find the

9 distance between two points in a coordinate system. Determine whether a set of three numbers could form a triangle by using the Triangle Inequality Theorem, and then determine whether that triangle is a right triangle using the Converse of the Pythagorean Theorem. Understand a proof of the Converse of the Pythagorean Theorem. Use the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in three dimensions. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real world and mathematical problems. Distinguish between categorical data and numerical data. Distinguish bivariate data from univariate data. Construct two-way tables of bivariate data. Interpret two-way tables of bivariate data. Calculate frequencies and relative frequencies in sets of bivariate data. Use percent s to express frequencies and relative frequencies. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

10 Unit 1 Concepts of Mathematics Scope and Sequence Unit 2 Scientific Notation & the Laws of Exponents 5 days 12 days 12 days Unit Description: Students review essential prerequisite learning s from previous years Unit Objectives: Use mathematical vocabulary from previous grades. Use all numerical operations with rational numbers Locate rational numbers on the number line, or ordered pairs on the coordinate plane. Solve one-and two-step equations in one variable. Create tables, graphs, and equations for simple relations. Unit Description: Students will represent large and small numbers with Scientific Notation, then extend their understanding to Laws of Exponents Unit Objectives: Use numbers expressed in the form of a single digit times an integer power of 10 (scientific notation) to represent very large or very small quantities. Convert numbers of any form to scientific notation. Apply scientific notation to information about Global Issues or Science to enable comparisons. Express how many times as much one number in scientific notation is than another. Express how much larger or smaller (as a difference) one number in scientific notation is than another. 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 Interpret scientific notation that has been generated by technology. Use zero and negative exponents in scientific notation before abstract study of exponent properties has begun. Know and apply the properties of integer exponents to generate equivalent numerical expressions, generalized from properties noted during the study of scientific notation. Use zero and negative exponents for any base, including bases, which are variables. Simplify expressions, which use exponents, especially those in scientific Unit 3 Functions and their Representations Unit Description: Students will understand functions and why they are interesting mathematically, representing them in several ways. Unit Objectives: Understand that a function is a rule that assigns to each input exactly one output. Understand that a function is a relation, and that most learning comes from observing relationships. Understand that a relation is a set of ordered pairs. Understand that there are nonmathematical functions, which are useful to consider. Understand that functions may be represented as tables, graphs, equations, sets of ordered pairs, verbal rules, and visual descriptions known as mappings. Understand that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Understand that relations may be represented as tables, graphs, equations, sets of ordered pairs, verbal rules, visual descriptions known as mappings, and understand the quality (one-to-one correspondence) which makes a relation a function. Switch among the various representations of functions, especially for comparing their properties. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables,

11 notation, but also for numerical expressions in other bases and abstract algebraic expressions. Understand and apply the Order of Operations when simplifying numerical or algebraic expressions. or by verbal descriptions). Use function notation.

12 Unit 4 Analyzing Linear Functions in Two Variables Unit 5 Solving Linear Equations in One Variable 23 days 5 days 12 days Unit Description: Students will be able to translate fluently from each of the representations of a linear function to the others, using concepts of slope and intercept, and they will be able to fit lines to real-world data sets for use in prediction. Unit Objectives: Represent proportional relationships as tables, graphs, and equations. Graph proportional relationships, interpreting the slope of the graph as the unit rate. Compare two different proportional relationships represented in different ways. Describe the graph of a non-vertical line through the origin using the equation y = mx or y = mx + 0. Describe the table of a horizontal or vertical line. Create a table of ordered pairs (x, y) for the graph of a line. Create a table of ordered pairs (x, y) for an equation in two variables. Create a graph for a linear equation in two variables. Create a graph for a table of ordered pairs. Write an equation for a linear graph. Write an equation for a table of ordered pairs from a linear relation. Distinguish tables of linear functions from tables of functions, which are not linear. Describe the graph of a non-vertical line intercepting the y-axis at b using the equation y = mx + b. Describe the graph of a horizontal or vertical line using the equations y = k or x = k. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Give examples of functions that are not Unit Description: Students will be able to solve linear equations in one variable using inverse operations, including equations involving distributive property, equations with variables on both sides, and literal equations, then represent real problems as solvable equations Unit Objectives: Solve linear equations with rational number coefficients in one variable by using inverse transformations. Understand that linear equations may have one, zero, or infinitely many solutions, and recognize which sequence of transformations will lead to which result. Solve linear equations, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve problems with linear equations in one variable. Unit 6 Solving Systems of Linear Equations Unit Description: Students will be able to solve systems of two linear equations graphically and algebraically, then represent real problems as solvable systems. Unit Objectives: Analyze and solve pairs of simultaneous linear equations. Graph a system of two linear equations on the same coordinate plane, and understand that solution to a system of two linear equations in two variables corresponds to points of intersection of their graphs. Estimate the solution to a system of two linear equations by graphing. Understand that the graphed solution is only an estimate without testing it, and understand the need for algebraic methods to get exact solutions for systems. Solve simple systems of two linear equations in two variables by inspection. Solve a system of two linear equations in two variables by substitution. Solve a system of two linear equations in two variables by elimination, or linear combination. Solve real world and mathematical problems leading to two linear equations in two variables. Understand that a system of linear

13 linear. Understand slope as rate of change, constant of proportionality, unit rate. Understand, use, and apply the slope formula. 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. Construct a function to model a linear relationship between two quantities. Determine the rate of change (slope) and initial value of the function (y-intercept) 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 (slope) and initial value of the function (y-intercept) of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Use the vocabulary of mathematical functions: increasing, decreasing, linear, and nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 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. Know that straight lines are widely used to model relationships between two quantitative variables. Informally fit a straight line (trend line, or line of best fit) to scatter plots that suggest a linear association. Assess the linear model s fit by judging the closeness of the data points to the line. Use the equation of a linear model to make predictions about bivariate measurements. Use the equation of a linear model to solve problems, interpreting the slope and y- intercept appropriately in the context of bivariate measurement data. equations in two variables may have one, zero, or infinitely many solutions, and understand how this is comparable to the number of solutions for a linear equation of one variable.

14 Unit 7 Congruence and Similarity Transformations Unit 8 Pythagorean Theorem and Measurement Unit 9 Other Topics from Geometry 27 days 15 days 4 days Unit Description: Students will define congruence and similarity in terms of transformations, understanding the mechanics of transformation with or without the coordinate plane. Unit Description: Students will understand the Pythagorean Theorem as a relationship among squares drawn on sides of a right triangle, then use it to solve for missing Unit Description: Students will understand facts about angle relationships that follow from transformations, and use various formulas about volumes of solid shapes Unit Objectives: Verify experimentally the properties of rotations, reflections, and translations. Use the properties of rigid transformations: lines are taken to lines, parallel lines to parallel lines, line segments to line segments of the same length, angles to angles of the same measure. 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 of rigid transformations that maps one to the other. Describe the effect of dilations, translations, rotations, and reflections on coordinates of figures. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rigid transformations or dilations. Given two similar two-dimensional figures, describe a sequence that maps one to the other. Use the vocabulary of geometry for transformations and angle relations. measurements Unit Objectives: Understand a proof of the Pythagorean Theorem, which uses sums of square areas. Use the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two dimensions. Understand how the Pythagorean Theorem can be used to solve problems by adding imaginary lines to make right triangles on drawings of problems. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Determine whether a set of three numbers could form a triangle by using the Triangle Inequality Theorem, then determine whether that triangle is a right triangle using the Converse of the Pythagorean Theorem. Understand a proof of the Converse of the Pythagorean Theorem. Unit Objectives: Use arguments following from properties of transformations 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. Use the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in three dimensions. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real world and mathematical problems.

15 Unit 10 Rational and Irrational Numbers Unit 11 Frequency Tables in Statistics 15 days 5 days 20 days Unit Description: Students will understand whether numbers are rational or irrational, and why most roots are irrational. Unit Description: Students will understand how predictions and hypotheses are tested with relative frequency. Unit Objectives: Write the decimal expansion of any fraction by using long division, and understand that the decimal expansion found by long division will repeat or end. Write any terminating or repeating decimal as a fraction, using their knowledge of place values for terminating decimals, or an algebraic algorithm for repeating decimals. Know that numbers that are not rational are called irrational. Know that irrational numbers are non-terminating, non-repeating decimals with no fraction representation. Know that pi and most square and cube roots are irrational. Use decimal approximations of irrational numbers to compare them to each other, as decimals on the number line. Use decimal approximations of irrational numbers to operate with them. 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, especially those equations generated by the Pythagorean Theorem. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Unit Objectives: Distinguish between categorical data and numerical data. Distinguish bivariate data from univariate data. Construct two-way tables of bivariate data. Interpret two-way tables of bivariate data. Calculate frequencies and relative frequencies in sets of bivariate data. Use percent s to express frequencies and relative frequencies. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. Unit 12 Pre-Algebra Unit Description: Students will understand foundational concepts for next year s Algebra class. Unit Objectives: Analyze non-linear functions

16 Unit Description Unit Title Concepts of Mathematics 8 Unit Summary Students review essential prerequisite learning from previous years, 5 days Learning Objectives Based on Mastery Skills Use mathematical vocabulary from previous grades. Use all numerical operations with rational numbers Locate rational numbers on the number line, or ordered pairs on the coordinate plane. Solve one-and two-step equations in one variable. Create tables, graphs, and equations for simple relations. Essential Questions Why is learning mathematics important? How do we use numbers to understand the world? How do we apply the basic properties and rules of real numbers? How can you communicate mathematical ideas effectively? Evidence of Learning (Assessment) Unit quiz Repeated demonstration of understanding as learning is extended Quarter 1 Summative Assessment will test mastery of these skills Required Lesson Activities Create tables, graphs, equations for simple relations Create drawings using coordinates in all four quadrants Demonstrate understanding of numerical operations on all types of rational numbers without using a calculator (in context of making tables and solving equations) Solve one- and two-step equations in one variable Add, Subtract, Multiply, Divide Number Pairs Resources Geogebra Equation Solving Cartesian Cartoons Glencoe Prealgebra Chapter 4 Glencoe Algebra Chapter 2 ConnectED (online resource provided with Glencoe Prealgebra, Algebra)

17 Unit Description Unit Title Scientific Notation & the Laws of Exponents Unit Summary Students will represent large and small numbers with Scientific Notation, then extend their understanding to Laws of Exponents, 12 days Learning Objectives Based on Mastery Skills Use numbers expressed in the form of a single digit times an integer power of 10 (scientific notation) to represent very large or very small quantities. Convert numbers of any form to scientific notation. Apply scientific notation to information about Global Issues or Science to enable comparisons. Express how many times as much one number in scientific notation is than another. Express how much larger or smaller (as a difference) one number in scientific notation is than another. 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 Interpret scientific notation that has been generated by technology. Use zero and negative exponents in scientific notation before abstract study of exponent properties has begun. Know and apply the properties of integer exponents to generate equivalent numerical expressions, generalized from properties noted during the study of scientific notation. Use zero and negative exponents for any base, including bases which are variables. Simplify expressions, which use exponents, especially those in scientific notation, but also for numerical expressions in other bases and abstract algebraic expressions. Understand and apply the Order of Operations when simplifying numerical or algebraic expressions. Essential Questions How should we think about things, which involve very large or very small numbers? Why are numbers represented in many ways? How is scientific notation used? How are exponents useful in the real world? Why is it useful to simplify expressions? Evidence of Learning (Assessment) World Populations Project: collect, order, compare, operate on data Small Numbers Project: collect, order, compare, operate on data Benchmark Assessment Unit 2 quizzes to assess mastery of scientific notation to standard form conversion, mastery of laws of exponents Quarter 1 Summative Assessment Ready Common Core Unit 1 assessment Required Lesson Activities Data Around Us Investigation 3: Dialing Digits, Ordering Rounded Numbers World Populations Project: collect, order, compare, operate on data Powers of Ten video Place values and Powers of Ten

18 Operations with Scientific Notation Laws of Exponents Small Numbers Project: collect, order, compare, operate on data Resources Glencoe Pre-Algebra Chapter 9 Glencoe Algebra Chapter 7 Ready Common Core Mathematics 8 Unit 1 EngageNY.org Mathematics 8 Module 1 ConnectED (online resource provided with Glencoe Pre-Algebra, Algebra) Connected Mathematics: Data around Us

19 Unit Description Unit Title Functions and their Representations Unit Summary Students will understand functions and why they are interesting mathematically, representing them in several ways, 12 days Learning Objectives Based on Mastery Skills Understand that a function is a rule that assigns to each input exactly one output. Understand that a function is a relation, and that most learning comes from observing relationships. Understand that a relation is a set of ordered pairs. Understand that there are non-mathematical functions, which are useful to consider. Understand that functions may be represented as tables, graphs, equations, and sets of ordered pairs, verbal rules, and visual descriptions known as mappings. Understand that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Understand that relations may be represented as tables, graphs, equations, sets of ordered pairs, verbal rules, visual descriptions known as mappings, and understand the quality (one-to-one correspondence), which makes a relation a function. Switch among the various representations of functions, especially for comparing their properties. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Give examples of functions that are not linear. Essential Questions How can we model relationships between quantities? How are patterns used when comparing two quantities? How can you communicate mathematical ideas effectively? How can you find and use patterns to model real-world situations? Evidence of Learning (Assessment) Benchmark Assessment Unit 3 quizzes to assess mastery of distinguishing functions from relations, multiple models of relations and functions Quarter 1 Summative Assessment Ready Common Core Unit 2 assessment Required Lesson Activities Relationships: one-to-one, many-to-one, one-to-many Many models of relationships Function Vocabulary Function Representations Connected Mathematics: Thinking with Mathematical Models activities for non-linear functions Desmos.com: Function Carnival Resources Glencoe Pre-Algebra Chapter 1, 8 Glencoe Algebra Chapter 1 Ready Common Core Mathematics 8 Unit 2, Lessons 6, 7 EngageNY.org Mathematics 8 Module 5

20 ConnectED (online resource provided with Glencoe Pre-Algebra, Algebra) Connected Mathematics: Thinking with Mathematical Models Investigation 4 Desmos.com: Function Carnival

21 Unit Description Unit Title Analyzing Linear Functions in Two Variables Unit Summary Students will be able to translate fluently from each of the representations of a linear function to the others, using concepts of slope and intercept, and they will be able to fit lines to real-world data sets for use in prediction, 23 days Learning Objectives Based on Mastery Skills Represent proportional relationships as tables, graphs, and equations. Graph proportional relationships, interpreting the slope of the graph as the unit rate. Compare two different proportional relationships represented in different ways. Describe the graph of a non-vertical line through the origin using the equation y = mx or y = mx + 0. Describe the table of a horizontal or vertical line. Create a table of ordered pairs (x, y) for the graph of a line. Create a table of ordered pairs (x, y) for an equation in two variables. Create a graph for a linear equation in two variables. Create a graph for a table of ordered pairs. Write an equation for a linear graph. Write an equation for a table of ordered pairs from a linear relation. Distinguish tables of linear functions from tables of functions, which are not linear. Describe the graph of a non-vertical line intercepting the y-axis at b using the equation y = mx + b. Describe the graph of a horizontal or vertical line using the equations y = k or x = k. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Understand slope as rate of change, constant of proportionality, unit rate. Understand, use, and apply the slope formula. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane. Construct a function to model a linear relationship between two quantities. Determine the rate of change (slope) and initial value of the function (y-intercept) 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 (slope) and initial value of the function (y-intercept) of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Use the vocabulary of mathematical functions: increasing, decreasing, linear, and nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 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. Know that straight lines are widely used to model relationships between two quantitative variables. Informally fit a straight line (trend line, or line of best fit) to scatter plots that suggest a linear association. Assess the linear model s fit by judging the closeness of the data points to the line. Use the equation of a linear model to make predictions about bivariate measurements. Use the equation of a linear model to solve problems, interpreting the slope and y-intercept appropriately in the context of bivariate measurement data.

22 Essential Questions Why are lines useful as representations of real-world ideas? How may lines be understood mathematically? How is change understood mathematically? How are the representations of lines (sequences, tables, graphs, equations) related? How is mathematics used to make predictions? Evidence of Learning (Assessment) Benchmark Assessment Unit 4 quizzes to assess mastery of representing linear relationships, mastery of slope and y-intercept concepts Green Globs Assessment (if available) Project: Drawing from Equations Quarter 2 Summative Assessment Ready Common Core Unit 2 and 3 assessments Project: Trendline from Data Set Required Lesson Activities Generate data from real-life experiments, tabulate and graph the data Moving Straight Ahead: Investigations 1, 2 Thinking with Mathematical Models, Investigations 1, 2 Create tables, graphs from equations Create equations, graphs from tables Create tables, equations from graphs Write slopes and intercepts from tables Write slopes and intercepts from graphs Write slopes and intercepts from equations Write slopes (rates of change) and intercepts (starting points) from equations Green Globs: represent graphs as equations Project: Drawing from Equations Find y-intercept given two points (solve for b) Draw lines of best fit onto scatterplots Write equations for lines of best fit, make predictions from equations Resources Glencoe Pre-Algebra Chapter 8 Glencoe Algebra Chapter 3 Ready Common Core Mathematics 8 Unit 2, Lessons 8-10; Unit 3 Lessons 11-13; Unit 5 Lessons EngageNY.org Mathematics 8 Module 4 ConnectED (online resource provided with Glencoe Pre-Algebra, Algebra) Connected Mathematics: Moving Straight Ahead Investigation 1, 2 Connected Mathematics: Thinking with Mathematical Models Investigation 1, 2 Desmos.com or Green Globs

23 Solving Linear Equations in One Variable Unit Description Unit Title Unit Summary Students will be able to solve linear equations in one variable using inverse operations, including equations involving distributive property, equations with variables on both sides, and literal equations, then represent real problems as solvable equations, 5 days Learning Objectives Based on Mastery Skills Solve linear equations with rational number coefficients in one variable by using inverse transformations. Understand that linear equations may have one, zero, or infinitely many solutions, and recognize which sequence of transformations will lead to which result. Solve linear equations, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Solve problems with linear equations in one variable. Essential Questions Why is it useful to model situations with numbers? How can situations be modeled with algebra? How can algebra help us to understand the situations? Why is it useful to simplify real situations and their models? What does it mean to say that one thing is equal to another? Evidence of Learning (Assessment) Benchmark Assessment Geogebra Equation Solving Activities Unit 5 quiz to assess mastery of solving equations Quarter 2 Summative Assessment Ready Common Core Unit 3 assessment Solving Equations with Rational Coefficients Solving Equations with Variables on Both Sides Solving Equations with Distributive Property Solving Literal Equations Geogebra Equation Solving Activities Required Lesson Activities Resources Glencoe Prealgebra Chapter 4, 5 Glencoe Algebra Chapter 2 Geogebra Equation Solving Activities Ready Common Core Mathematics 8 Unit 3 Lesson 14 EngageNY.org Mathematics 8 Module 4 ConnectED (online resource provided with Glencoe Prealgebra, Algebra)

24 Unit Description Unit Title Solving Systems of Linear Equations Unit Summary Students will be able to solve systems of two linear equations graphically and algebraically, then represent real problems as solvable systems, 12 days Learning Objectives Based on Mastery Skills Analyze and solve pairs of simultaneous linear equations. Graph a system of two linear equations on the same coordinate plane, and understand that solution to a system of two linear equations in two variables corresponds to points of intersection of their graphs. Estimate the solution to a system of two linear equations by graphing. Understand that the graphed solution is only an estimate without testing it, and understand the need for algebraic methods to get exact solutions for systems. Solve simple systems of two linear equations in two variables by inspection. Solve a system of two linear equations in two variables by substitution. Solve a system of two linear equations in two variables by elimination, or linear combination. Solve real world and mathematical problems leading to two linear equations in two variables. Understand that a system of linear equations in two variables may have one, zero, or infinitely many solutions, and understand how this is comparable to the number of solutions for a linear equation of one variable. Essential Questions Why should a concept be modeled as a line? What are the implications of a linear model when considering a concept? What does it mean when two lines intersect? What does it mean to solve an equation? What does it mean to solve a system of equations? Evidence of Learning (Assessment) Benchmark Assessment Geogebra Equation Solving Activities Unit 6 quizzes to assess mastery of solving systems Quarter 2 Summative Assessment Ready Common Core Unit 3 assessment Required Lesson Activities Solving systems by graphing Solve systems by substitution Solve systems by elimination Solve word problems using systems Resources Glencoe Pre-Algebra Chapter 8 Glencoe Algebra Chapter 6 Dolciani Algebra Worksheets, Text Connected Math 2: The Shapes of Algebra Ready Common Core Mathematics 8 Unit 3 Lesson EngageNY.org Mathematics 8 Module 4 ConnectED (online resource provided with Glencoe Pre-Algebra, Algebra)

25 Congruence and Similarity Transformations Unit Description Unit Title Unit Summary Students will define congruence and similarity in terms of transformations, understanding the mechanics of transformation with or without the coordinate plane, 27 days Learning Objectives Based on Mastery Skills Verify experimentally the properties of rotations, reflections, and translations. Use the properties of rigid transformations: lines are taken to lines, parallel lines to parallel lines, line segments to line segments of the same length, angles to angles of the same measure. 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 of rigid transformations that maps one to the other. Describe the effect of dilations, translations, rotations, and reflections on coordinates of figures. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rigid transformations or dilations. Given two similar two-dimensional figures, describe a sequence that maps one to the other. Use the vocabulary of geometry for transformations and angle relations. Essential Questions How is motion understood mathematically? How do we communicate about motion? How is transformation related to motion? Why do we need to think about shapes when we think about motion? What is the difference between congruence and similarity? Do all transformations maintain congruence or similarity? Do transformations guarantee symmetry? Evidence of Learning (Assessment) Benchmark Assessment Unit 7 quizzes to assess mastery of transforming shapes, generating sequences of transformations, writing coordinates of transformations, defining congruence and similarity Quarter 3 Summative Assessment Ready Common Core Unit 4 assessment Required Lesson Activities EngageNY Module 2 Lessons 1-5, compress 6-11 into three days, (12 days on rigid transformation) Geogebra Coordinate Transformation Activities Glencoe Transformations Maze EngageNY Module 3 Lessons 1-3 in two days, Lessons 4, 5, 7, 8 in two days, Lessons 9-12 in two days (7 days on dilation and similarity) Project: create a transformation maze Glencoe Pre-Algebra Chapter 2, 6 Glencoe Geometry Chapter 10 Geogebra Coordinate Transformation Activities Resources