Grade 8UNIT 4: Linear Equationssuggested Number of Days for Entire UNIT:41

Grade 8UNIT 4: Linear EquationsSuggested Number of Days for Entire UNIT:41

Essential Question / Key Concepts / Cross Curricular Connections
What strategies can be used to model and solve linear situations?
Vocabulary
Slope
System of Linear Equations
Solution to a system of linear equations
Linear Expression
Equation
Unit rate
Term
Coefficient
Like terms
Variable
Solution /

Writing and Solving Linear Equations
Linear Equations in Two Variables and Their Graphs
Slope and Equations of Lines
Systems of Linear Equations and Their Solutions
Pythagorean Theorem (optional)

*Assessment and Review
Mid-Module Assessment and Review: After Section B (5 days, included in Unit Instructional Days)
End-of-Module Assessment and Review: After Section D (5 days, included in Unit Instructional Days) / Science: Physics concept of speed, acceleration and distance.
Unit Outcome (Focus)
Students extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations. Students learn to apply the skills they acquired in Grades 6 and 7, with respect to symbolic notation and properties of equality to transcribe and solve equations in one variable and then in two variables.

UNIT 4SECTION A: Writing and Solving Linear Equations Suggested Number of Days for SECTION: 9

Essential Question / Key Concepts / Standards for Mathematical Practice
What is the connection between proportional relationships, lines and linear equations? /

Writing Equations Using Symbols
Linear and Non-Linear Expressions in ?
Linear Equations in ?
Solving a Linear Equation
Writing and Solving Linear Equations
Solutions of a Linear Equation
Classification of Solutions
Linear Equations in Disguise
An Application of Linear Equations.

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UNIT 4 SECTION B: Linear Equations in Two Variables and Their Graphs Suggested Number of Days for SECTION:5

Essential Question / Key Concept / Standards for Mathematical Practice
How can we graph proportional relationships?? /

A Critical Look at Proportional Relationships
Constant Rate
Linear Equations in Two Variables
The Graph of a Linear Equation in Two Variables
The Graph of a Linear Equation―Horizontal and Vertical Lines.

/ 6. Attend to precision
7. Look for and make use of structure
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Students work with constant speed, a concept learned in Grade 6, but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane (8.EE.C.8a). Students begin to investigate the shape of a graph of a linear equation. Also in this topic is the standard form of a linear equation, ??+??=?, and when ?,?≠0, a non-vertical line is produced. Further, when ? or ?=0, then a vertical or horizontal line is produced. / 8.EE.5
(DOK 2) / Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. / 

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UNIT: 4SECTION: C:Slope and Equations of LinesSuggested Number of Days for SECTION:9

Essential Question / Key Concept / Standards for Mathematical Practice
How can you graph proportional relationships? /

The Slope of a Non-Vertical Line
The Computation of the Slope of a Non-Vertical Line
The Line Joining Two Distinct Points of the Graph ? = ?? + ? has Slope ?
There is Only One Line Passing Through a Given Point with a Given Slope
The Graph of a Linear Equation in Two Variables is a Line
Every Line is a Graph of a Linear Equation
Some Facts about Graphs of a Linear Equation in Two Variables Constant Rates Revisited
The Defining Equation of a Line

/ 1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
7. Look for and make use of structure
Comments
Students know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph (8.EE.B.5). In this topic, students derive ?=?? and ?=??+? for linear equations by examining similar triangles. Students generate graphs of linear equations in two variables first by completing a table of solutions, then using information about slope and ?-intercept. Students learn some basic facts about lines and equations, such as why two lines with the same slope and a common point are the same line, how to write equations of lines given slope and a point, and how to write an equation given two points. Students compare two different proportional relationships represented by graphs, tables, equations, or descriptions. Finally, students learn that multiple forms of an equation can define the same line. / Standard No.
8.EE.B.5
(DOK 2)
8.EE.B.6
(DOK 2) / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways
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 ?=?? for a line through the origin and the equation ?=??+b of a line intercepting the vertical axis at b / Priority



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UNIT 4SECTION D: Systems of Linear Equations and Their Solutions Suggested Number of Days for SECTION: 7

Essential Question / Key Concepts / Standards for Mathematical Practice
What is the connection between proportional relationships, lines and linear equations? /

Introduction to Simultaneous Equations
Geometric Interpretation of the Solutions of a Linear System Characterization of Parallel Lines
Nature of Solutions of a System of Linear Equations
Another Computational Method of Solving a Linear System
Word Problems
Conversion Between Celsius and Fahrenheit

/ 1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
4. Model with mathematics
7. Look for and make use of structure
8.Look for and express regularity in repeated reasoning
Comments / Standard No. / Standard
 Major Standard Supporting Standard Additional Standard
 Standard ends at this grade Fluency Standard / Priority
Simultaneous equations and their solutions are the focus of Section D. Students graph simultaneous linear equations to find the point of intersection and then verify that the point of intersection is in fact a solution to each equation in the system (8.EE.C.8a). Students graph systems of linear equations whose solutions do not have integer coordinates. Students use an estimation of the solution from the graph to verify their algebraic solution is correct. Students learn to solve systems of linear equations by substitution and elimination (8.EE.C.8b). Students understand that a system can have a unique solution, no solution, or infinitely many solutions, as they did with linear equations in one variable. Finally, students apply their knowledge of systems to solve problems in real-world contexts, including converting temperatures from Celsius to Fahrenheit / 8.EE.B.5
8.EE.B.6 / Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways
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.

Resources/Possible Activities

PREDICT HOW MANY SOLUTIONS A LINEAR EQUATION HAS In this lesson you will learn to predict how many solutions a linear equation has by identifying key features of equations.

SYSTEMS OF EQUATIONS GAME: Solve different systems of equations and have fun in the same time when playing this interactive game. This game can be played alone or with another student or it can be full class by dividing the classroom in two teams. Interactive games can be found online at and select Algebra Gamesfrom left menu. Scroll down until yousee Systems of Equations Game.

JUMP IT OUT – RATE OF CHANGE EXPERIMENT: Conduct a short experiment to model a linear relationship and practice logging the data into a table then graphing the data. Have students time themselves for 1 – 2 minutes doing jumping-jacks (or sit-ups). Have a student record time in pre-determined increments (every 10 seconds), another person calling out the number of jumping-jacks or sit-ups in that time increment, and a third person recording the data. After collecting the data, have students create a t-chart. Look for a steady pattern. Teach students how to use the algorithm {y2 – y1/ x2 – x1} to compute rate of change. Have students set up a graph, determine which variables go on which axis, set appropriate scale, and plot coordinate points from the graph. Teach students how to find rate of change from a graph. Use the method above to find slope. Discuss how to compare information given in each form (graph vs. table). Extend: Project a table of data and a graph that have different slopes. Discuss and compare the slopes of each set ofdata. Plot a point and travel the slope, to graph a line at Click on the Algebra –6-8 square and click Line Plotter.

MATCH-UP! SLOPE-INTERCEPT TO STANDARD FORMS AND VICE VERSA: Create linear equations in the slope-intercept form and also their standard form. Write them on half sheets of paper, tape different papers to students, and have them find their match.

CHANGING COST PER MINUTE: In this lesson, one of a multi-part unit from Illuminations, students use an interactive graph to explore the relationship between change and accumulation. In this activity, the cost per minute for phone use changes after the first sixty minutes of calls. Students analyze how changing the cost per minute shown in the first graph affects the total cost shown in the second graph.

FROM PATTERNS OF INPUT AND OUTPUT TO ALGEBRAIC EQUATIONS: Students explore the relationship between input and output values and learn to use algebraic expressions and equations.

MATH JOURNALS: WRITING AND SOLVING LINEAR EQUATIONS IN STANDARD FORM: Have students solve and explain various linear problems in their math journals. Ex: You have $24 to buy party supplies for a class party. Juice costs $3 per big bottle, and chips cost $2 per big bag. Write an equation in standard form that relates the amount of juice and chips you can buy, by using the $24. How many different chip/juice combinations can you have? (Answer: 3x + 2y = 24)Linear equation and systems of equations practice drills with answer key can be found at: Click on Algebra and pick any of the many and varied practice worksheets.

SIMPLIFY AND SOLVE LINEAR EQUATIONS ACTIVITY: Using wooden 1" cubes, create “dice” (you can also cover the sides of a traditional die with stickers or masking tape). Two dice should have numbers (both single and double digits); one should have operations (addition, subtraction multiplication, division); one should have variables. Students should roll the dice and create equation. Students should then arrange the equation into slope intercept form. Have students check one another’s work.

SYSTEM OF EQUATIONS STATIONS: Create stations of equation examples on tables throughout the room. Group students and have them travel around the room “art walk” style to each station. Have students answer the questions in their math journals. (Note: In a system of equations a solution point satisfies both equations simultaneously.) Students can solve by graphing or algebraically by setting equal to each other and solving for x. Ex: Suppose you have $20 in your savings account and you start saving $5 every week. Your sister has $5 in her savings account and starts saving $10 everyweek. Neither of you make any withdrawals or extra deposits. After how many weeks will you have the same amount of money in your accounts? (Answer: 3 weeks). How much money will this be? (Answer: $35). Systems of equations can also be explored online at Clickon Ages 11–16 at left, click on Simultaneous Systems of EquationsGraphs and/or Algebra.

CONCENTRATION MATCH-UP: Create index cards and half sheets of graph paper with linear equations in standard form and their graphs. Flip them over. Have students play as they would play Concentration. Students can use x- and y-intercepts as a quick mental way to visualize each line’s graph and find the intersection point or they can solve the system of equations algebraically to find the intersection point

EQUATION STORIES: Project a story problem, and discuss it with the class. Identify which term depends on a variable, and which does not. Project four linear equations and have students match the correct one to the story. Ex 1: A toy company spends $1500 a day for factory expenses plus $8 to make each teddy bear. The bears sell for $12 each. Which of the following equations can be used to find the number of teddy bears the company has to sell in one day to equal its daily costs? a) 1500 + 8t = 12 b) 12 + 8t = 1500 c) 1500 + 8t = 12t d) 8t = 12t + 1500. Additional problems can be found at:

Click on Algebra at top and pick any of the many practice worksheets.

SOLUTIONS – ONE, NONE, INFINITELY And MANY: Create linear equations, some of which are identical, others of which have the same slope but differing y-intercepts. Have students choose pairs of equations and algebraically manipulate them to find which are identical (infinitely many solutions), or have identical slope and different y-intercept (no solutions). Those with different slope and y-intercept have one solution.

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