Mathematics 8 (Grade 7/8)
3. Analyzing Functions and Equations (8.F/8.EE)
In this unit, students will grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
Students will also use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or xcoordinate changes by an amount A, the output or ycoordinate changes by the amount mA. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and yintercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.
Students will also use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or xcoordinate changes by an amount A, the output or ycoordinate changes by the amount mA. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and yintercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.
What should my child know and be able to do?
Students will:
Define, evaluate, and compare functions.
8.F.A.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.A.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.
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.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 distancetime graph to a distancetime equation to determine which of two moving objects has greater speed.
8.EE.B.6
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; 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.
Define, evaluate, and compare functions.
8.F.A.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.
Use functions to model relationships between quantities.
8.F.B.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 (SAT® Content  HOA.04  HOA.06).
8.F.B.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.
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7
Solve linear equations in one variable (SAT® Content  HOA.01  HOA.03).
8.EE.C.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.C.7.B
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations (SAT® Content  HOA.07  HOA.08).
8.EE.C.8.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 (SAT® Content  HOA.07  HOA.08).
8.EE.C.8.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 (SAT® Content  HOA.07  HOA.08).
8.EE.C.8.C
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.
Students will:
Define, evaluate, and compare functions.
8.F.A.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.
 Check for Understanding: Recognize Functions from Tables  Recognize Functions from Graphs
 Review/Rewind: Checking if Two Quantities Represents a Function: Height
 Enrichment Tasks: Foxes and Rabbits  Garbage, Variation 1
8.F.A.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.
 Check for Understanding: Compare Linear Functions  Comparing Linear Functions Word Problem
 Review/Rewind: Comparing Linear Functions  Comparing Linear Functions Word Problem: Climb
 Enrichment Tasks: Battery Charging
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.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 distancetime graph to a distancetime equation to determine which of two moving objects has greater speed.
 Check for Understanding: Graphing Proportional Relationships  Rates and Proportional Relationships
 Review/Rewind: Rates and Proportional Relationships: Gas Mileage
 Enrichment Tasks: Coffee by the Pound  Stuffing Envelopes
8.EE.B.6
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; 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.
 Review/Rewind: Proving Slope is Constant
 Enrichment Tasks: Slopes Between Points on a Line
Define, evaluate, and compare functions.
8.F.A.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.
 Check for Understanding: Linear and NonLinear Functions
 Review/Rewind: Recognizing Linear Functions
 Enrichment Tasks: Introduction to Linear Functions
Use functions to model relationships between quantities.
8.F.B.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 (SAT® Content  HOA.04  HOA.06).
 Check for Understanding: Linear Models Word Problems  Solutions to 2Variable Equations  Slope from Graph  Interpreting Slope and yintercept of Lines of Best Fit
 Review/Rewind: Linear Functions Word Problem: Iceberg
 Enrichment Tasks: Video Streaming  High School Graduation
8.F.B.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.
 Check for Understanding: Graphing Linear Functions Word Problems  Interpreting Graphs of Functions
 Review/Rewind: Interpreting a Graph Example
 Enrichment Tasks: Tides  Bike Race
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7
Solve linear equations in one variable (SAT® Content  HOA.01  HOA.03).
 Check for Understanding: Equations with Variables on Both Sides  Equations with Parentheses
 Review/Rewind: Ex. 1: Distributive Property to Simplify
 Enrichment Tasks: The Sign of Solutions  Solving Equations
8.EE.C.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).
 Check for Understanding: Number of Solutions to Equations  Number of Solutions to Equations Challenge
 Review/Rewind: Number of Solutions to Equations Examples
8.EE.C.7.B
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
 Check for Understanding: Equation Practice with Angle Addition  Equation Practice with Segment Addition  Equation Practice with Midpoints  Equation Practice with Vertical Angles  Equations Practice with Variables on Both Sides  MultiStep Equations with Distribution
 Review/Rewind: Ex. 2: Distributive Property to Simplify
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations (SAT® Content  HOA.07  HOA.08).
 Check for Understanding: Age Word Problems  Solve Linear Systems of Equations  Systems of Equations with Elimination Challenge  Systems of Equations with Elimination  Systems of Equations with Substitution  Systems of Equations Word Problems: Shelves  Systems of Equations Word Problems
 Review/Rewind: Systems of Equations with Substitution: Potato Chips
 Enrichment Tasks: Cell Phone Plans  Kimi and Jordan
8.EE.C.8.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 (SAT® Content  HOA.07  HOA.08).
 Check for Understanding: Systems of Equations with Graphing  Number of Solutions to a System of Equations Graphically
 Review/Rewind: How Do You Solve a System of Equations by Graphing?
 Enrichment Tasks: The Intersection of Two Lines
8.EE.C.8.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 (SAT® Content  HOA.07  HOA.08).
 Check for Understanding: Number of Solutions to a System of Equations Algebraically  Solve Linear Systems of Equations  Systems of Equations with Elimination Challenge  Systems of Equations with Elimination  Systems of Equations with Substitution  Systems of Equations Word Problems
 Review/Rewind: How to Determine the Number of Solutions of a System of Equations using Algebraic Reasoning (Example)
8.EE.C.8.C
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.
 Check for Understanding: Age Word Problems  Systems of Equations Word Problems
 Review/Rewind: How to Solve a Word Problem with a System of Equations (Example)
 Enrichment Tasks: Quinoa Pasta 1  Summer Swimming
What are some signs of student mastery?

Tools & Technology
Desmos is a free online graphing calculator that works on any computer or tablet without requiring any downloads. A FREE Desmos iPad app is available too! Pan Balance  Expressions (NCTM) Students are able to represent each side of an equation on a balance with a given value of x. Using properties of equality, students can explore and utilize the balance in a variety of ways. Watch as these students use their bodies as tools to graph linear equations (8.EE.B.5). 
What are functions and how can students represent functions in different ways (8.F.B.4)? View this classroom video to learn more.
Watch this teacher as he explains the mathematics behind solving a system of equations (8.EE.C.8).
