Horizontal Progression Recommendations

Greater Albany Public Schools 8 th Grade Math CCSS Pacing Guide Horizontal Progression Recommendations 4 2 Solving Equations 8EE7a-b, 8NS1, 8EE2 5 3 Triangles & Angles with Pythagorean Theorem 6 (5) (end of Sem 1) 8 (to spring break) 4 8G5, 8G6, 8G7, 8G8 4 Linear Equations & Slope 8F4, 8EE5, 8EE6, 8F1, 8F5, 8F3 5 Systems of Linear Equations 8F2, 8EE8a-c 6 Statistics 8SP1, 8SP2, 8SP3, 8SP4 OR8.2.1, OR 8.2.2, OR8.2.3, OR8.2.4, OR8.2.5, OR8.2.6, OR8.2.7, OR8.2.8 4 7 Coordinate Plane Geometry 8G1a-c, 8G2, 8G3, 8G4 2 8 Volume 8G9 Time frame Unit # Unit Title Unit Standards Notes 3 1 Number Systems 8EE1, 8NS2, 8EE3, 8EE4 Rationale: Beginning the school year with Unit 1 will offer toolswithout variables- for the students to build upon throughout the school year. We will continue to Solving Equations, a unit of review from 7 th grade and foundations for the topics that follow. Unit 3 allows students to take equations of degrees greater than 1 and put them into practice, for example, using Pythagorean Theorem to solve for a missing side of a triangle. The right triangles and triangle properties from Unit 3 lead right into slope triangles in Unit 4. Students will continue to work with visual displays of first-degree equations. Unit 5 extends Unit 4 by comparing two linear functions on the same coordinate plane. Unit 6 allows students the opportunity to extend linear equations into lines of best fit. The scatter plots are not always aligned in exact lines, so students will have to give reason as to equations that best represent the data. Unit 7 allows students to give geometrical names to the transitions of the systems of linear equations and how they might have translated or rotated a line to get the other line. Additionally, They can use dilations to relate slope triangles to explain the ratio of changing vertical distance to changing horizontal distance. This unit can tie each of the previous units together. The year will conclude with Unit 8, where students can explore three dimensional circular solids. Total = 36 wks GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 1

Greater Albany Public Schools 8 th Grade Math CCSS Pacing Guide Semester 1 Unit 1: Number Systems Time Frame: 3 weeks 8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 3 5 = 3 3 = 1/3 3 = 1/27. 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. For example, estimate the population of the United States as 3 10 8 and the population of the world as 7 10 9, and determine that the world population is more than 20 times larger. 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. Big Ideas The structure of the base 10 systems allows us to represent and perform operations with very large and very small quantities. How are very large or very small numbers represented? What are efficient strategies for manipulating very large/very small numbers? Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources Scientific Notation can be used to represent very large Combine numbers with like bases using and very small numbers. exponent rules. Positive integer exponents represent repeated multiplication. Non-zero bases raised to a zero power is 1. Negative integer exponents represent repeated division. Power to power (example) Add exponents on like base (example) Simplify expressions with scientific notation using properties of integer exponents. Choose appropriate units of measurement within a given context. Interpret scientific notation that has been generated on technology. Convert between standard & scientific notation Exponent Scientific notation Base Integers Irrational numbers Truncate Power Base 10 Place value Decimal GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 2

Semester 1 Unit 2: Solving Equations Time Frame: 4 weeks 8.EE.7. Solve linear equations in one variable. 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). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 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., 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1and 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 2 is irrational. GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 3

Big Ideas All linear equations with rational number coefficients and one variable will result in one real solution, no real solutions or all real numbers as a solution. Square roots and cube roots are used to solve quadratic and cubic equations, respectively. What are the possible solutions to linear equations in one variable? How can equations in degrees higher than 1 be solved? How are the numbers in the real number system categorized? The numbers in the real number system are categorized by their common properties, allowing for some sets of numbers to be subsets of another number set. Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources Understand square roots/cube roots Solve multi-step linear equations in 1 variable with rational coefficients and variables on each side. Understand number of solutions and real world example for each situation. Square roots of non-perfect squares are irrational. Real numbers that are not rational are called irrational. Every number has a decimal expansion; rational numbers have a decimal expansion that eventually repeats. Give examples of one solution, no real solutions or all real solutions. Solve equations of the form x 2 =p and x 3 =p. Expand expressions using the distributive property. Rational numbers Irrational numbers Real numbers Linear equations Solutions Perfect squares Square roots Cubes Cube roots Coefficient Even exponents will have two roots and odd exponents will have 1 root. Understand situations exist that have one solution, no real solutions or all real solutions. Collect like terms to solve equations. Convert a repeating decimal into a rational number. Identify perfect squares and perfect cubes. Evaluate square roots and cube roots of small perfect squares and perfect cubes. Approximate, compare and locate real (including irrationals) numbers. GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 4

Semester 1 Unit 3: Triangles & Angles w/pythagorean Theorem Time Frame: 5 weeks 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. 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.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. Big Ideas Proof is a series of logical arguments that uses one or more assumed truths to get to the next argument. The Pythagorean Theorem and its Converse can be used to find distances between any two points and create right angles. What is a mathematical proof? How is the Pythagorean Theorem used in real-world? GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 5

I Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources 2 If a 2 +b 2 =c is true for a set of side lengths, where c is the longest side, then the set form a right triangle. Only sets of side lengths that satisfy the Pythagorean Theorem form right triangles. Only parallel lines cut by a transversal create congruent or supplementary angles relationships. The sum of the areas of two squares with side lengths a and b will equal the area of the square with side length c. By creating a right triangle, Pythagorean Theorem can be used to find distance between two points. If two of the angles in two different triangles are congruent, the two triangles are similar. An exterior angle of a triangle is supplementary to the adjacent interior angle. Use the Pythagorean Theorem to find missing sides of a right triangle in two- and threedimensional figures. Use the Pythagorean Theorem to determine if a triangle is a right triangle. Use the Pythagorean Theorem to find distance between two points. Create a right triangle from any two given points. Given two angles in a triangle, find the third angle. Use proportional relationships and knowledge about similar triangles to find missing side lengths. Given two interior angles of a triangle, find all the exterior angles. Create an argument for each angle relationship in an image where two parallel lines cut by a transversal. Pythagorean Theorem Interior angle Exterior angle Parallel line Transversal Similarity Congruence Interior angle sum Converse of Pythagorean Theorem Hypotenuse Leg Right angle Obtuse angle Acute angle Coordinate plane x-axis y-axis Adjacent angle Central angle Complementary angle Supplementary angle Isosceles triangle Scalene triangle Vertical angles Corresponding angles Alternate interior angle Alternate exterior angle Same side interior GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 6

Semester 1 Unit 4: Linear Equations & Slope Time Frame: 6 weeks (to end of Sem 1) 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.EE.5. 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.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.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. (Function notation is not required in Grade 8.) 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.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. 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. Big Ideas Functions represent a pairing of one input to a unique output within a context. Representing functions using graphs, tables, words, and/or equations can help to analyze a given situation. How can a function be represented? What situations can be represented by a function? What is slope? Slope explains how the values in a functional situation are changing. GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 7

Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources Equations are linear relationships only if there is a Construct a function to model a linear constant rate of change. relationship between two values (x, y). A function assigns each input to one output. Slope is the ratio of the vertical change to the horizontal change. Graphing the solutions of y=mx+b will result in only a straight line. Creating a table from y=mx+b or a verbal representation of a linear relationship will demonstrate a constant rate of change. Determine rate of change and initial value from a table, graph, words, or equation. Interpret the rate of change and initial value for a given situation. Graph proportional relationships. Compare two different proportional relationships in different ways. Interpret unit rate in terms of the slope of the graph. Use similar triangles to prove the slope is the equivalent between any two points on the same line. Derive the equations y= mx and y=mx+b. Sketch a graph that models a verbal description. Give examples of functions that are nonlinear. Find the slope of a line. Linear function Linear equation Rate of change Slope Ordered pair Input/output table Graph Unit rate Direct variation Similar triangle Slope triangles Coordinate plane Derive Origin Vertical axis Horizontal axis Increasing Decreasing Non-linear Linear Parabola Hyperbola Exponential Rise/Run Qualitative features Proportional relationships Initial value y-intercept x-intercept Undefined slope No slope Slope-intercept form Standard form Positive slope Negative slope GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 8

Semester 2 Unit 5: Systems of Linear Equations Time Frame: 8 weeks (to Spring Break) 8.F.2. 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.EE.8. Analyze and solve pairs of simultaneous linear equations. a. 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. b. 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. c. 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 GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 9

Big Ideas Systems of linear functions can have common points, called solutions, even though the functions represent different relationships of the same variables. Linear equations can be represented using tables, graphs, various forms of equations, words. Each of these representations can be manipulated to be any other form so that the two functions can be analyzed. How can systems of linear equations be used to solve real-world problems? How can information in different forms be manipulated to analyze systems of equations? Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources The solution(s) in a system of linear equations defines the Compare properties of two functions. point(s) where the two linear functions cross. Two linear functions can cross one time, be the same line, or be parallel to each other. Two linear equations create a system only if their variables represent the same information. Identify the number of solutions for a given set of functions. Solve pairs of simultaneous linear equations algebraically (substitution or elimination) to find exact solutions. Estimate solutions by graphing. Given two pairs of points, determine the number of solutions in the system. Apply systems of linear equations to solve real-world problems. Solve simple systems by inspection. Convert between various forms of linear equations to analyze the system. System of linear equations Intersection Substitution Elimination GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 10

Semester 2 Unit 6: Statistics Time Frame: 4 weeks 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. 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.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? GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 11

Big Ideas Relationships can be drawn between two sets of data. Frequently, the relationships can be modeled using linear equations. Data can be modeled using various displays. Each display allows for drawing patterns to make inferences about the data. How do linear equations help with modeling and using real world data? How do different displays of data help to analyze the information collected? Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources Straight lines are used to model relationships. Construct and interpret scatter plots. Scatter plot Outlier The closer a trend line is to all the data, the better model of the relationship it will be. Investigate and describe patterns of association between two quantities. Bivariate data Positive association Negative association Patterns of association can be seen in bivariate categorical data. Approximate trend lines. Cluster Linear association Informally assess the fit of a trend line. Non-linear association Line of best fit Solve problems using linear models (trend lines). Trend line Linear model Categorical data Create a linear model from a trend line. Quantitative data Frequency Construct and interpret a two-way table. Relative frequency Two-way table Calculate relative frequency from a two-way table. Rows Columns Describe possible associations between two variables. GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 12

Semester 2 Unit 7: Coordinate Plane Geometry 8.G.1. 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. Time Frame: 4 weeks 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. 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. Big Ideas Any translation, reflection, rotation applied to a pre-image will result in a congruent image. How do transformations relate a pre-image to the image? A dilation applied to a pre-image s center will result in a similar image. Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources A two-dimensional figure is congruent to another if it can Describe the effect of transformations. be obtained from the first using a sequence translations, rotations and/or reflections. A two-dimensional figure is similar to another if it can be obtained from the first using a sequence transformations. Describe a sequence of transformations that exhibits similarity or congruence between figures. Rotate, reflect, translate and/or dilate shapes and determine if the resulting image is similar or congruent. Describe the effect of transformations using coordinates. Rotations Transformation Translation Reflections Dilation Line segment Similarity Congruency Line Angle Parallel Sequence Lines of symmetry Pre-image image GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 13

Semester 2 Unit 8: Volume Time Frame: 2 weeks 8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Big Ideas Formulas are proven relationships. Given partial information, formulas can be used to find the missing information. How are formulas used to solve real-world and mathematical problems? Concepts: (Students will know) Skills: (Students will be able to) Unit Vocabulary Resources Know the formulas for volume of spheres, cones and cylinders. Use the formulas for volume of spheres, cones and cylinders to solve problems. Solve cubic equations. Find cube root of a number. Volume Sphere Cone Cylinders Radius Diameter Circle Base Pi Circumference Height Slant height Pythagorean Theorem Cubic equations Dimension GAPS MS Math Pacing Guide 8 th Grade 2013-14 DRAFT 14