PreAlgebra GT (Grade 6)
3: Analyzing Functions and Linear Equations (7.RP/8.EE/8.F)
Students 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 or y = mx) 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 (m)(A). 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 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.
What will my child learn?
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.
Analyze proportional relationships and use them to solve realworld and mathematical problems.
7.RP.A.2
Recognize and represent proportional relationships between quantities.
7.RP.A.2.A
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.A.2.B
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.C
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.D
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
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.
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.
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.
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
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.
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.
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: Recognizing Functions  Determine the Domain of Functions  Recognize Functions from Tables
 Review/Rewind: Checking if Two Quantities Represent a Function
 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: Comparing Linear Functions  Comparing Linear Functions Applications
 Review/Rewind: Comparing Linear Functions
 Enrichment Tasks: Battery Charging
Analyze proportional relationships and use them to solve realworld and mathematical problems.
7.RP.A.2
Recognize and represent proportional relationships between quantities.
 Check for Understanding: Analyzing and Identifying Proportional Relationships
 Review/Rewind: Intro to Proportional Relationships
 Enrichment Tasks: Art Class, Assessment Variation  Climbing the steps of El Castillo
7.RP.A.2.A
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
 Check for Understanding: Analyzing and Identifying Proportional Relationships  Proportional Relationships: Graphs
 Review/Rewind: Proportional Relationships: Graphs
 Enrichment Tasks: Buying Bananas, Assessment Variation  Art Class, Variation 1*
7.RP.A.2.B
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
 Check for Understanding: Rate Problems 1
 Review/Rewind: Equations of Proportional Relationships
 Enrichment Tasks: Buying Coffee*  Walkathon 2
7.RP.A.2.C
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
 Check for Understanding: Analyzing and Identifying Proportional Relationships  Writing Proportional Equations
 Review/Rewind: Writing Proportional Equations
 Enrichment Tasks: Art Class Variation 2*  Proportionality
7.RP.A.2.D
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
 Check for Understanding: Analyzing and Identifying Proportional Relationships  Interpreting Graphs of Proportional Relationships
 Review/Rewind: Interpreting Graphs of Proportional Relationships
 Enrichment Tasks: Robot Races*  Robot Races, Assessment Variation
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: Graphing Proportional Relationships: Unit Rate, Rates and Proportional Relationships
 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.
 Check for Understanding: Slope and Triangle Similarity
 Review/Rewind: Finding Slope From a Graph  Graph from SlopeIntercept Equation
 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.
 Check for Understanding: Constructing and Interpreting Linear Functions  Equations From Tables  Linear Equations Word Problems  Slope from Graph  Equation of a Line: Any Form  Interpreting Linear Functions  Linear Functions Intercepts  Slope Intercept Equation from Two Points  Solving for the xIntercept  Solving Intercept Equation from Graph
 Review/Rewind: SlopeIntercept Form Problems
 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: Constructing and Interpreting Linear Functions  Increasing and Decreasing Intervals  Interpreting Graphs of Linear and NonLinear Functions  Interpreting Graphs of Functions
 Review/Rewind: How to Match Features of a Modeling Function to their Real World Meaning
 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.
 Check for Understanding: Equations with Variables on Both Sides  Linear Equations with One, Zero, or Infinite Solutions  MultiStep Equations with Distribution
 Review/Rewind: Why do we do things to both sides: Variables on Both Sides
 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: Linear Equations with One, Zero, or Infinite Solutions
 Review/Rewind: Number of Solutions to Linear Equations
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  Integer Sums  MultiStep Equations with Distribution
 Review/Rewind: Equations with Variables on Both Sides: Fractions
7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
 Check for Understanding: Proportions 1  Constructing Proportions to Solve Application Problems  Discount, Tax, and Tip Word Problems  Markup and Commission Word Problems  Rate Problems 2  Writing Proportions
 Review/Rewind: Solving Percent Problems  Ratio Word Problems: Boys to Girls
 Enrichment Tasks: The Price of Bread  TwoSchool Dance
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
 Check for Understanding: Age Word Problems  Graphing Systems of Equations  Systems of Equations  Systems of Equations with Elimination  Systems of Equations with Simple Elimination  Systems of Equations with Substitution  Systems of Equations Word Problems  Test Solutions to Systems of Equations  Understanding Systems of Equations Word Problems
 Review/Rewind: Systems of Equations: Trolls, Tolls
 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.
 Check for Understanding: Graphing Systems of Equations  Systems of Equations  Graphing Systems of Equations with One, Zero, or Infinite Solutions  Understanding Systems of Equations Word Problems
 Review/Rewind: Number of Solutions to Systems of Equations
 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.
 Check for Understanding: Constructing Consistent and Inconsistent Systems  Systems of Equations  Systems of Equations with Elimination  Systems of Equations with Simple Elimination  Systems of Equations with Substitution  Understanding Systems of Equations Word Problems
 Review/Rewind: Systems of Equations: Substitution  Systems of Equations: Elimination  Systems of Equations: Graphing 
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  Understanding Systems of Equations Word Problems
 Review/Rewind: How to Solve a Word Problems with Systems of Equations
 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! Algebra vs The Cockroaches (Hotmath) Determine the equation of the line the cochroaches are going and blast them away! Algebra tiles: Use tiles to represent variables and constants, learn how to represent and solve algebra problems, and solve equations. Solve a system of equations by graphing (IXL) Practice graphing linear equations finding their solution. Watch as these students use their bodies as tools to graph linear equations (8.EE.B.5). 
More 4 U
Have you ever heard your child talk about using Algebra Tiles in math class? Watch these videos that demonstrate how Algebra Tiles can help students understand the mathematics behind: 