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STUDENT EDITION Grade 8 VOLUME 2 Mission 4 Linear Equations and Linear Systems Mission 5 Functions and Volume Mission 6 Associations in Data NAME

2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum used under the CC BY 4 0 license Download the original for free at openupresources org Zearn is a registered trademark Printed in the U S A ISBN 979 8 88868 882 3

Table of Contents Mission 4 Lesson 1 Number Puzzles 3 Lesson 2 Keeping the Equation Balanced 9 Lesson 3 Balanced Moves 13 Lesson 4 More Balanced Moves 19 Lesson 5 Solving Any Linear Equation 25 Lesson 6 Strategic Solving 31 Lesson 7 All Some or No Solutions 37 Lesson 8 How Many Solutions 43 Lesson 9 When Are They the Same 47 Lesson 10 On or Off the Line 53 Lesson 11 On Both of the Lines 59 Lesson 12 Systems of Equations 65 Lesson 13 Solving Systems of Equations 71 Lesson 14 Solving More Systems 77 Lesson 15 Writing Systems of Equations 83 Lesson 16 Solving Problems with Systems of Equations 89 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license iii

Mission 5 iv Lesson 1 Inputs and Outputs 93 Lesson 2 Introduction to Functions 99 Lesson 3 Equations for Functions 105 Lesson 4 Tables Equations and Graphs of Functions 111 Lesson 5 More Graphs of Functions 119 Lesson 6 Even More Graphs of Functions 127 Lesson 7 Connecting Representations of Functions 133 Lesson 8 Linear Functions 139 Lesson 9 Linear Models 145 Lesson 10 Piecewise Linear Functions 151 Lesson 11 Filling Containers 157 Lesson 12 How Much Will Fit 163 Lesson 13 The Volume of a Cylinder 169 Lesson 14 Finding Cylinder Dimensions 175 Lesson 15 The Volume of a Cone 181 Lesson 16 Finding Cone Dimensions 187 Lesson 17 Scaling One Dimension 193 Lesson 18 Scaling Two Dimensions 201 Lesson 19 Estimating a Hemisphere 207 Lesson 20 The Volume of a Sphere 215 Lesson 21 Cylinders Cones and Spheres 221 Lesson 22 Volume As a Function of 227 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Mission 6 Lesson 1 Organizing Data 233 Lesson 2 Plotting Data 239 Lesson 3 What a Point in a Scatter Plot Means 245 Lesson 4 Fitting a Line to Data 253 Lesson 5 Describing Trends in Scatter Plots 261 Lesson 6 The Slope of a Fitted Line 269 Lesson 7 Observing More Patterns in Scatter Plots 277 Lesson 8 Analyzing Bivariate Data 283 Lesson 9 Looking for Associations 289 Lesson 10 Using Data Displays to Find Associations 297 Lesson 11 Gone in 30 seconds 305 Lesson 12 Keeping Track of All Possible Outcomes 307 Lesson 13 Multi step Experiments 315 Terminology 321 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license v

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Grade 8 Mission 4 Linear Equations and Linear Systems

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ZEARN MATH STUDENT EDITION G8M4 LESSON 1 Lesson 1 Number Puzzles Let s solve some puzzles Warm Up 1 What do you notice What do you wonder 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Concept Exploration ACTIVITY 1 2 Solve each puzzle Show your thinking Organize it so it can be followed by others 1 The temperature was very cold Then the temperature doubled Then the temperature dropped by 10 degrees Then the temperature increased by 40 degrees The temperature is now 16 degrees What was the starting temperature 2 Lin ran twice as far as Diego Diego ran 300 m farther than Jada Jada ran 31 the distance that Noah ran Noah ran 1200 m How far did Lin run 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 3

G8M4 LESSON 1 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Write another number puzzle with at least three steps On a different piece of paper write a solution to your puzzle Trade puzzles with your partner and solve theirs Make sure to show your thinking With your partner compare your solutions to each puzzle Did they solve them the same way you did Be prepared to share with the class which solution strategy you like best 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 1 Lesson Summary Here is an example of a puzzle problem Twice a number plus 4 is 18 What is the number There are many different ways to represent and solve puzzle problems We can reason through it Twice a number plus 4 is 18 Then twice the number is 18 4 14 That means the number is 7 We can draw a diagram We can write and solve an equation x x x 4 18 x 14 x 7 2x 4 18 2x 14 x 7 Reasoning and diagrams help us see what is going on and why the answer is what it is But as number puzzles and story problems get more complex those methods get harder and equations get more and more helpful We will use different kinds of diagrams to help us understand problems and strategies in future lessons but we will also see the power of writing and solving equations to answer increasingly more complex mathematical problems 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 5

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ZEARN MATH STUDENT EDITION G8M4 LESSON 1 Name Date GRADE 8 MISSION 4 LESSON 1 Exit Ticket Andre and Elena are reading the same book over the summer Andre says he has read 51 of the book Elena says she has read 20 more pages than Andre If Elena is on page 55 how many pages are in the book Lin has drawn a diagram to solve this question Find her error a 20 55 55 55 55 55 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 7

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ZEARN MATH STUDENT EDITION G8M4 LESSON 2 Lesson 2 Keeping the Equation Balanced Let s figure out unknown weights on balanced hangers Warm Up 1 What do you notice What do you wonder Concept Exploration ACTIVITY 1 2 This picture represents a hanger that is balanced because the weight on both sides is the same Answer the questions about the hanger below 1 Elena takes two triangles off of the left side and three triangles off of the right side Will the hanger still be in balance or will it tip to one side Which side Explain how you know 2 If a triangle weighs 1 gram how much does a square weigh 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 9

G8M4 LESSON 2 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 In these two hangers a triangle weights 3 grams and a circle weights 6 grams Answer the questions about these hangers below 1 Find the weight of a square in Hanger A and the weight of a pentagon in Hanger B 2 Write an equation to represent each hanger A B Lesson Summary If we have equal weights on the ends of a hanger then the hanger will be in balance If there is more weight on one side than the other the hanger will tilt to the heavier side We can think of a balanced hanger as a metaphor for an equation An equation says that the expressions on each side have equal value just like a balanced hanger has equal weights on each side If we have a balanced hanger and add or remove the same amount of weight from each side the result will still be in balance We can do these moves with equations as well adding or subtracting the same amount from each side of an equation maintains the equality a 2b 5b a 3b 10 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M4 LESSON 2 Date GRADE 8 MISSION 4 LESSON 2 Exit Ticket Here is a hanger that is in balance We don t know how much any of its shapes weigh How could you change the number of shapes on it but keep it in balance Describe in words or draw a new diagram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 11

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ZEARN MATH STUDENT EDITION G8M4 LESSON 3 Lesson 3 Balanced Moves Let s rewrite equations while keeping the same solutions Warm Up 1 Figures A B C and D show the result of simplifying the hanger in Figure A by removing equal weights from each side A B C D Here are some equations Each equation represents one of the hanger diagrams 2 x 3y 4x 2y 2y x 2 x 3y 2z 2z 4x 2y x 3y 2x y 1 Write the equation that goes with each figure a b c d 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 13

G8M4 LESSON 3 ZEARN MATH STUDENT EDITION 2 Each variable x y and z represents the weight of one shape Which goes with which 3 Explain what was done to each equation to create the next equation If you get stuck think about how the hangers changed Concept Exploration ACTIVITY 1 2 14 You will receive some cards Each of the cards 1 through 6 show two equations Each of the cards A through E describe a move that turns one equation into another 1 Match each number card with a letter card 2 One of the letter cards will not have a match For this card write two equations showing the described move 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 3 ACTIVITY 2 3 Noah and Lin both solved the equation 14a 2 a 3 Do you agree with either of them Why Noah s solution Lin s solution 14a 2 a 3 14a 2 a 3 14a 2a 6 7a a 3 12a 6 6a 3 a 1 2 a 12 43 Elena is asked to solve 15 10x 5 x 9 What do you recommend she does to each side first 53 Diego is asked to solve 3x 8 4 x 5 What do you recommend he does to each side first 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 15

G8M4 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary An equation tells us that two expressions have equal value For example if 4x 9 and 2x 3 have equal value we can write the equation 4x 9 2x 3 Earlier we used hangers to understand that if we add the same positive number to each side of the equation the sides will still have equal value It also works if we add negative numbers For example we can add 9 to each side of the equation 4x 9 9 2x 3 9 4x 2x 12 add 9 to each side combine like terms Since expressions represent numbers we can also add expressions to each side of an equation For example we can add 2x to each side and still maintain equality 4x 2x 2x 12 2x 6x 12 add 2x to each side combine like terms If we multiply or divide the expressions on each side of an equation by the same number we will also maintain the equality so long as we do not divide by zero 6x 61 12 16 or 6x 6 12 6 multiply each side by 16 divide each side by 6 Now we can see that x 2 is the solution to our equation We will use these moves in systematic ways to solve equations in future lessons 16 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 3 Name Date GRADE 8 MISSION 4 LESSON 3 Exit Ticket Match these equation balancing steps with the description of what was done in each step Step 1 12x 6 10 6x 3 5 Step 2 6x 3 5 6x 8 Step 3 6x 8 x 43 Descriptions to match with each step a Add 3 to both sides b Multiply both sides by 16 c Divide both sides by 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 17

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ZEARN MATH STUDENT EDITION G8M4 LESSON 4 Lesson 4 More Balanced Moves Let s rewrite some more equations while keeping the same solutions Warm Up 1 Which of these have the same solution as Equation 1 Be prepared to explain your reasoning Equation 1 x 3 2 4x Equation A Equation B Equation C Equation D 2x 6 4 8x x 5 4x 2 1 2x x 3 3 2 5x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 19

G8M4 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Two students are solving the same equation Answer the questions below Here is an equation and then all the steps Clare wrote to solve it Here is the same equation and the steps Lin wrote to solve it 14x 2x 3 3 5x 9 12x 3 3 5x 9 3 4x 1 3 5x 9 4x 1 5x 9 1 x 9 8 x 14x 2x 3 3 5x 9 12x 3 3 5x 9 12x 3 15x 27 12x 15x 24 3x 24 x 8 1 Are both of their solutions correct Explain your reasoning 2 Describe some ways the steps they took are alike and different 3 Mai and Noah also solved the equation but some of their steps have errors Find the incorrect step in each solution and explain why it is incorrect Mai Noah 14x 2x 3 3 5x 9 12x 3 3 5x 9 7x 3 3 9 7x 3 27 7x 24 x 24 7 20 14x 2x 3 3 5x 9 12x 3 15x 27 27x 3 27 27x 24 x 24 27 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 4 ACTIVITY 2 Solve these equations for x 43 1 12 6x 5 9 3 2 2 x 4 13 6x 54 3 3x 12 9x 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 21

G8M4 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary How do we make sure the solution we find for an equation is correct Accidentally adding when we meant to subtract missing a negative when we distribute forgetting to write an x from one line to the next there are many possible mistakes to watch out for Fortunately each step we take solving an equation results in a new equation with the same solution as the original This means we can check our work by substituting the value of the solution into the original equation For example say we solve the following equation 2x 3 x 5 2x 3x 15 5x 15 x 3 Substituting 3 in place of x into the original equation 2 3 3 3 5 6 3 8 6 24 we get a statement that isn t true This tells us we must have made a mistake somewhere Checking our original steps carefully we made a mistake when distributing 3 Fixing it we now have 2x 3 x 5 2x 3x 15 5x 15 x 3 Substituting 3 in place of x into the original equation to make sure we didn t make another mistake 2 3 3 3 5 6 3 2 6 6 This equation is true so x 3 is the solution 22 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M4 LESSON 4 Date GRADE 8 MISSION 4 LESSON 4 Exit Ticket Lin solved the equation 8 x 3 7 2x 4 17 incorrectly Find the errors in her solution What should her answer have been Lin s solution 8 x 3 7 2x 4 17 8 x 3 7 2x 13 8x 24 7 26x 8x 17 26x 17 34x 1 2 x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 23

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ZEARN MATH STUDENT EDITION G8M4 LESSON 5 Lesson 5 Solving Any Linear Equation Let s solve linear equations Warm Up 1 Solve each equation mentally 1 5 x 8 2 1 x 2 3 3x 9 4 10 5x Concept Exploration ACTIVITY 1 2 You will get 4 cards each with an equation 1 With your partner select a card and choose who will take the first turn 2 During your turn decide what the next move to solve the equation should be explain your choice to your partner and then write it down once you both agree Switch roles for the next move This continues until the equation is solved 3 Choose a second equation to solve in the same way trading the card back and forth after each move 4 For the last two equations choose one each to solve and then trade with your partner when you finish to check one another s work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 25

G8M4 LESSON 5 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Tyler says he invented a number puzzle He asks Clare to pick a number and then asks her to do the following Triple the number Subtract 7 Double the result Subtract 22 Divide by 6 Clare says she now has a 3 Tyler says her original number must have been a 3 How did Tyler know that Explain or show your reasoning Be prepared to share your reasoning with the class 26 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 5 Lesson Summary When we have an equation in one variable there are many different ways to solve it We generally want to make moves that get us closer to an equation like variable some number For example x 5 or t 73 Since there are many ways to do this it helps to choose moves that leave fewer terms or factors If we have an equation like 3t 5 7 adding 5 to each side will leave us with fewer terms The equation then becomes 3t 2 Dividing each side of this equation by 3 will leave us with t by itself on the left and that t 23 Or if we have an equation like 4 5 a 12 dividing each side by 4 will leave us with fewer factors on the left 5 a 3 Some people use the following steps to solve a linear equation in one variable 1 Use the distributive property so that all the expressions no longer have parentheses 2 Collect like terms on each side of the equation 3 Add or subtract an expression so that there is a variable on just one side 4 Add or subtract an expression so that there is just a number on the other side 5 Multiply or divide by a number so that you have an equation that looks like variable some number For example suppose we want to solve 9 2b 6 3 b 5 4b 9 2b 6 3b 15 4b Use the distributive property 15 2b b 15 Gather like terms 15 3b 15 Add 2b to each side 30 3b Add 15 to each side 10 b Divide each side by 3 Following these steps will always work although it may not be the most efficient method From lots of experience we learn when to use different approaches 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 27

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ZEARN MATH STUDENT EDITION G8M4 LESSON 5 Name Date GRADE 8 MISSION 4 LESSON 5 Exit Ticket 1 14 Noah wanted to check his solution of x 14 5 for the equation 2 7x 6 6x 10 Substituting 5 for x he writes the following 1 14 14 2 7 5 6 6 5 10 14 7 14 5 6 12 5 20 14 5 7 14 5 6 5 12 5 20 7 14 6 12 14 20 98 6 168 20 92 148 Find the incorrect step in Noah s work and explain why it is incorrect 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 29

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ZEARN MATH STUDENT EDITION G8M4 LESSON 6 Lesson 6 Strategic Solving Let s solve linear equations like a boss Warm Up 1 1 The triangle and the square have equal perimeters Find the value of x 2x 2 2x What is the perimeter of each of the figures x 8 x 2 Concept Exploration ACTIVITY 1 2 Without solving identify whether these equations have a solution that is positive negative or zero 1 x 3x 6 4 5 9 4x 4 2 7x 3 25 6 8 5x 20 3 7x 32 5 7 12 8 5x 20 4 3x 11 11 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 31

G8M4 LESSON 6 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Use the equations to answer the questions below Here are a lot of equations a 56 8 5b 75 53 b f 3 c 1 2 3c 1 3c 1 b 12 t 3 10 6 5 g 4m 3 9 4m 4 8 c 10 v 2 v 17 4 d 2 4k 3 13 2 18 k 13 e 32 n 12 5n 5 7 h p 5 p 4 p 8 p i 2 2q 1 5 18 q j 2r 49 8 r 5 1 Without solving identify 3 equations that you think would be least difficult to solve and 3 equations you think would be most difficult to solve Be prepared to explain your reasoning 2 Choose 3 equations to solve At least one should be from your least difficult list and one should be from your most difficult list 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 6 Lesson Summary Sometimes we are asked to solve equations with a lot of things going on on each side For example x 2 x 5 3 2x6 20 This equation has variables on each side parentheses and even a fraction to think about Before we start distributing let s take a closer look at the fraction on the right side The expression 2x 20 is being multiplied by 3 and divided by 6 which is the same as just dividing by 2 so we can re write the equation as x 2 x 5 2x 2 20 But now it s easier to see that all the terms on the numerator of the right side are divisible by 2 which means we can re write the right side again as x 2 x 5 x 10 At this point we could do some distribution and then collect like terms on each side of the equation Another choice would be to use the structure of the equation Both the left and the right side have something being subtracted from x But if the two sides are equal that means the something being subtracted on each side must also be equal Thinking this way the equation can now be re written with less terms as 2 x 5 10 Only a few steps left But what can we tell about the solution to this problem right now Is it positive Negative Zero Well the 2 and the 5 multiplied together are 10 so that means the 2 and the x multiplied together cannot have a positive or a negative value Finishing the steps we have 2 x 5 10 x 5 5 x 0 Divide each side by 2 Subtract 5 from each side Neither positive nor negative Just as predicted 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 33

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ZEARN MATH STUDENT EDITION G8M4 LESSON 6 Name Date GRADE 8 MISSION 4 LESSON 6 Exit Ticket 1 Without solving identify whether this equation has a solution that is positive negative or zero 3x 5 3 2 Solve the equation x 5 x 1 x 2x 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 35

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ZEARN MATH STUDENT EDITION G8M4 LESSON 7 Lesson 7 All Some or No Solutions Let s think about how many solutions an equation can have Warm Up 1 Which one doesn t belong 1 5 7 7 5 2 5 7 7 5 3 2 7 5 4 5 7 7 5 Concept Exploration ACTIVITY 1 2 1 Solve the problems below Label these equations true for all values or true for no values n n 5 9 3x 10 6 3x 2t 6 2 t 3 1 x 1 x 2 3 3 n 1 3n 1 1 20d 4 5d 4 2 y 6 3 2 y 9 v 2 v 2 Write the other side of this equation so that this equation is true for all values of u 6 u 2 2 3 Write the other side of this equation so that this equation is true for no values of u 6 u 2 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 37

G8M4 LESSON 7 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 1 Solve the problems below Complete each equation so that it is true for all values of x a 3x 6 3 x b x 2 x 15x 10 5 2 c 2 Complete each equation so that it is true for no values of x a 3x 6 3 x b x 2 x 15x 10 5 2 c 3 38 Describe how you know whether an equation will be true for all values of x or true for no values of x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 7 Lesson Summary An equation is a statement that two expressions have an equal value The equation 2x 6 is a true statement if x is 3 2 3 6 It is a false statement if x is 4 2 4 6 The equation 2x 6 has one and only one solution because there is only one number 3 that you can double to get 6 Some equations are true no matter what the value of the variable is For example 2x x x is always true because if you double a number that will always be the same as adding the number to itself Equations like 2x x x have an infinite number of solutions We say it is true for all values of x Some equations have no solutions For example x x 1 has no solutions because no matter what the value of is it can t equal one more than itself When we solve an equation we are looking for the values of the variable that make the equation true When we try to solve the equation we make allowable moves assuming it has a solution Sometimes we make allowable moves and get an equation like this 8 7 This statement is false so it must be that the original equation had no solution at all 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 39

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ZEARN MATH STUDENT EDITION G8M4 LESSON 7 Name Date GRADE 8 MISSION 4 LESSON 7 Exit Ticket 3x 8 3x What value could you write in after 3x that would make the equation true for 1 no values of x 2 all values of x 3 just one value of x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 41

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ZEARN MATH STUDENT EDITION G8M4 LESSON 8 Lesson 8 How Many Solutions Let s solve equations with different numbers of solutions Warm Up 1 Consider the unfinished equation 12 x 3 18 Match the following expressions with the number of solutions the equation would have with that expression on the right hand side a 6 2x 3 1 One solution b 4 3x 3 2 No solutions c 3 All solutions 4 2x 3 Concept Exploration ACTIVITY 1 2 You will each get some cards Use them to solve the problems below 1 With your partner solve each equation 2 Then sort them into categories 3 Describe the defining characteristics of those categories and be prepared to share your reasoning with the class 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 43

G8M4 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary Sometimes it s possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions For example look at 2 12x 18 6 18x 6 x 7 Using the distributive property on the left and right sides we get 24x 36 6 18x 6x 42 From here combining like terms gives us 24x 42 24x 42 Since the left and right sides of the equation are the same we know that this equation is true for any value of x without doing any more moves Similarly we can sometimes use structure to tell if an equation has no solutions For example look at 6 6x 5 12 3x 2 12 If we think about each move as we go we can stop when we realize there is no solution 1 1 6 6 6x 5 6 12 3x 2 12 Multiply each side by 61 6x 5 2 3x 2 2 Distribute 16 on the right side 6x 5 6x 4 2 Distribute 2 on the right side The last move makes it clear that the constant terms on each side 5 and 4 2 are not the same Since adding 5 to an amount is always less than adding 4 2 to that same amount we know there are no solutions Doing moves to keep an equation balanced is a powerful part of solving equations but thinking about what the structure of an equation tells us about the number of solutions is just as important TERMINOLOGY constant term 44 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 8 Name Date GRADE 8 MISSION 4 LESSON 8 Exit Ticket Elena began to solve this equation 12x 6 4x 3 2 6x 4 2 3 12x 6 4x 3 3 2 6x 4 2 12x 6 4x 3 6 6x 4 6 12x 24x 18 36x 24 6 When she got to the last line she stopped and said the equation is true for all values of x How could Elena tell 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 45

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ZEARN MATH STUDENT EDITION G8M4 LESSON 9 Lesson 9 When Are They the Same Let s use equations to think about situations Warm Up 1 If you were babysitting which would you rather Explain your reasoning Charge 5 for the first hour and 8 for each additional hour Or Charge 15 for the first hour and 6 for each additional hour Concept Exploration ACTIVITY 1 2 1 2 The amount of water in two tanks every 5 minutes is shown in the table Describe what is happening in each tank Either draw a picture say it verbally or write a few sentences Use the table to estimate when the tanks will have the same amount of water Time minutes Tank 1 liters Tank 2 liters 0 25 1 000 5 175 900 10 325 800 15 475 700 20 625 600 25 775 500 30 925 400 35 1 075 300 40 1 225 200 45 1 375 100 50 1 525 0 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 47

G8M4 LESSON 9 3 ZEARN MATH STUDENT EDITION The amount of water in liters in tank 1 after t minutes is 30t 25 The amount of water in liters in tank 2 after t minutes is 20t 1 000 Find the time when the amount of water will be equal ACTIVITY 2 3 48 A building has two elevators that both go above and below ground At a certain time of day the travel time it takes elevator A to reach height h in meters is 0 8h 16 seconds The travel time it takes elevator B to reach height h in meters is 0 8h 12 seconds 1 What is the height of each elevator at this time 2 How long would it take each elevator to reach ground level at this time 3 If the two elevators travel toward one another at what height do they pass each other How long would it take 4 If you are on an underground parking level 14 meters below ground which elevator would reach you first 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 9 Lesson Summary Imagine a full 1 500 liter water tank that springs a leak losing 2 liters per minute We could represent the number of liters left in the tank with the expression 2x 1 500 where x represents the number of minutes the tank has been leaking Now imagine at the same time a second tank has 300 liters and is being filled at a rate of 6 liters per minute We could represent the amount of water in liters in this second tank with the expression 6x 300 where x represents the number of minutes that have passed Since one tank is losing water and the other is gaining water at some point they will have the same amount of water but when Asking when the two tanks have the same number of liters is the same as asking when 2x 1 500 the number of liters in the first tank after x minutes is equal to 6x 300 the number of liters in the second tank after x minutes 2x 1 500 6x 300 Solving for x gives us x 150 minutes So after 150 minutes the number of liters of the first tank is equal to the number of liters of the second tank But how much water is actually in each tank at that time Since both tanks have the same number of liters after 150 minutes we could substitute x 150 minutes into either expression Using the expression for the first tank we get 2 150 1 500 which is equal to 300 1 500 or 1 200 liters If we use the expression for the second tank we get 6 150 300 or just 900 300 which is also 1 200 liters That means that after 150 minutes each tank has 1 200 liters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 49

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ZEARN MATH STUDENT EDITION Name G8M4 LESSON 9 Date GRADE 8 MISSION 4 LESSON 9 Exit Ticket To own and operate a home printer it costs 100 for the printer and an additional 0 05 per page for ink To print out pages at an office store it costs 0 25 per page Let p represent number of pages 1 What does the equation 100 0 05p 0 25p represent 2 The solution to that equation is p 500 What does the solution mean 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 51

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ZEARN MATH STUDENT EDITION G8M4 LESSON 10 Lesson 10 On or Off the Line Let s interpret the meaning of points in a coordinate plane Warm Up 1 Which one doesn t belong Explain your reasoning A B y y x C y x D x y x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 53

G8M4 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Jada told Noah that she has 2 worth of quarters and dimes in her pocket and 17 coins all together She asked him to guess how many of each type of coin she has Here is a table that shows some combinations of quarters and dimes that are worth 2 Complete the table Number of quarters Number of dimes 0 20 4 0 5 2 Here is a graph of the relationship between the number of quarters and the number of dimes when there are a total of 17 coins 22 a What does Point A represent 20 b How much money in dollars is the combination represented by Point A worth number of dimes 18 16 14 12 10 A 8 6 4 2 2 4 6 8 10 12 14 16 18 20 22 24 number of quarters 54 3 Is it possible for Jada to have 4 quarters and 13 dimes in her pocket Explain how you know 4 How many quarters and dimes must Jada have Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 10 ACTIVITY 2 Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week Yesterday Andre made 15 signs and Clare made 5 signs Today they need to make more signs Each person s progress today is shown in the coordinate plane number of completed signs 3 D 60 50 B 40 30 20 A C 10 10 20 30 40 50 60 70 time in minutes 80 90 100 110 Based on the lines mark the statements as true or false for each person Point What it says A At 40 minutes I have 25 signs completed B At 75 minutes I have 42 and a half signs completed C At 0 minutes I have 15 signs completed D At 100 minutes I have 60 signs completed Clare Andre Lesson Summary We studied linear relationships in an earlier unit We learned that values of x and y that make an equation true correspond to points x y on the graph For example if we have x pounds of flour that costs 0 80 per pound and y pounds of sugar that costs 0 50 per pound and the total cost is 9 00 then we can write an equation like this to represent the relationship between x and y 0 8x 0 5y 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 55

G8M4 LESSON 10 ZEARN MATH STUDENT EDITION Since 5 pounds of flour costs 4 00 and 10 pounds of sugar costs 5 00 we know that x 5 y 10 is a solution to the equation and the point 5 10 is a point on the graph The line shown is the graph of the equation y 20 18 9 16 sugar pounds 16 14 12 1 14 5 10 10 8 6 4 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x flour pounds Notice that there are two points shown that are not on the line What do they mean in the context The point 1 14 means that there is 1 pound of flour and 14 pounds of sugar The total cost for this is 0 8 1 0 5 14 or 7 80 Since the cost is not 9 00 this point is not on the graph Likewise 9 pounds of flour and 16 pounds of sugar costs 0 8 9 0 5 16 or 15 20 so the other point is not on the graph either Suppose we also know that the flour and sugar together weigh 15 pounds That means that x y 15 If we draw the graph of this equation on the same coordinate plane we see it passes through two of the three labeled points y 20 18 9 16 sugar pounds 16 14 12 1 14 5 10 10 8 6 4 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x flour pounds The point 1 14 is on the graph of x y 15 because 1 14 15 Similarly 5 10 15 But 9 16 15 so 9 16 is not on the graph of x y 15 In general if we have two lines in the coordinate plane 56 The coordinates of a point that is on both lines makes both equations true The coordinates of a point on only one line makes only one equation true The coordinates of a point on neither line make both equations false 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 10 Name Date GRADE 8 MISSION 4 LESSON 10 Exit Ticket On the coordinate plane shown one line shows combinations of dimes and quarters that are worth 3 The other line shows combinations of dimes and quarters that total to 12 coins 18 number of dimes 16 dimes and quarters that total to 3 14 12 10 8 6 4 2 12 coins all together 2 4 6 8 10 12 14 16 18 20 number of quarters 1 Name one combination of 12 coins shown on the graph 2 Name one combination of coins shown on the graph that total to 3 3 How many quarters and dimes would you need to have both 12 coins and 3 at the same time 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 57

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ZEARN MATH STUDENT EDITION G8M4 LESSON 11 Lesson 11 On Both of the Lines Let s use lines to think about situations Warm Up 1 In the diagram below a lady bug and an ant start at a certain distance apart The lady bug s position is tracked above each number line The ant s position is tracked below each number line What do you notice What do you wonder ladybug start 0 seconds 2 seconds 4 seconds 6 seconds 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 59

G8M4 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 A different ant and ladybug are a certain distance apart and they start walking toward each other The graph shows the ladybug s distance from its starting point over time and the labeled point 2 5 10 indicates when the ant and the ladybug pass each other y 24 22 Distance centimeters 20 18 16 14 12 10 8 6 4 2 1 2 3 4 5 x Time seconds The ant is walking 2 centimeters per second 60 1 Write an equation representing the relationship between the ant s distance from the ladybug s starting point and the amount of time that has passed 2 If you haven t already draw the graph of your equation on the same coordinate plane 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 11 ACTIVITY 2 3 Elena and Jada were racing 100 meters on their bikes Both racers started at the same time and rode at constant speeds Here is a table that gives information about Jada s bike race Time from start seconds Distance from start meters 6 36 9 54 1 Graph the relationship between distance and time for Jada s bike race Make sure to label and scale the axes appropriately 2 Elena traveled the entire race at a steady 6 meters per second On the same set of axes graph the relationship between distance and time for Elena s bike race 3 Who won the race 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 61

G8M4 LESSON 11 ZEARN MATH STUDENT EDITION Lesson Summary The solutions to an equation correspond to points on its graph For example if Car A is traveling 75 miles per hour and passes a rest area when t 0 then the distance in miles it has traveled from the rest area after t hours is d 75t The point 2 150 is on the graph of this equation because 150 75 2 two hours after passing the rest area the car has traveled 150 miles If you have two equations you can ask whether there is an ordered pair that is a solution to both equations simultaneously For example if Car B is traveling towards the rest area and its distance from the rest area is d 14 65t we can ask if there is ever a time when the distance of Car A from the rest area is the same as the distance of Car B from the rest area If the answer is yes then the solution will correspond to a point that is on both lines y Distance miles 14 12 10 0 1 7 5 8 6 4 2 0 02 0 04 0 06 0 08 0 10 0 12 Time hours 0 14 0 16 0 18 0 20 0 22 x Looking at the coordinates of the intersection point we see that Car A and Car B will both be 7 5 miles from the rest area after 0 1 hours which is 6 minutes Now suppose another car Car C had also passed the rest stop at time t 0 and traveled in the same direction as Car A also going 75 miles per hour It s equation would also be d 75t Any solution to the equation for Car A would also be a solution for Car C and any solution to the equation for Car C would also be a solution for Car A The line for Car C would land right on top of the line for Car A In this case every point on the graphed line is a solution to both equations so that there are infinitely many solutions to the question when are Car A and Car C the same distance from the rest stop This would mean that Car A and Car C were side by side for their whole journey When we have two linear equations that are equivalent to each other like y 3x 2 and 2y 6x 4 we will get two lines that are right on top of each other Any solution to one equation is also a solution to the other so these two lines intersect at infinitely many points 62 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 11 Name Date GRADE 8 MISSION 4 LESSON 11 Exit Ticket Andre and Noah started tracking their savings at the same time Andre started with 15 and deposits 5 per week Noah started with 2 50 and deposits 7 50 per week The graph of Noah s savings is given and his equation is y 7 5x 2 5 where x represents the number of weeks and y represents his savings Write the equation for Andre s savings and graph it alongside Noah s What does the intersection point mean in this situation y 40 35 Savings dollars 30 25 20 15 10 5 1 2 3 4 5 6 Weeks 7 8 9 10 11 x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 63

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ZEARN MATH STUDENT EDITION G8M4 LESSON 12 Lesson 12 Systems of Equations Let s learn what a system of equations is Warm Up 1 Diego and Lin are drinking milkshakes Lin starts with 12 ounces and drinks 14 an ounce per second Diego starts with 20 ounces and drinks 23 an ounce per second Answer the questions about this situation below 1 How long will it take Lin and Diego to finish their milkshakes 2 Without graphing explain what the graphs in this situation would look like Think about slope intercepts axis labels units and intersection points to guide your thinking 3 Discuss your description with your partner If you disagree work to reach an agreement Concept Exploration ACTIVITY 1 2 There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake Han and Jada both decide to hike from the parking lot to the lake and back but they start their hikes at different times Answer the questions about their hikes below At the time that Han reaches the lake and starts to turn back Jada is 0 6 miles away from the parking lot and hiking at a constant speed of 3 2 miles per hour towards the lake Han s distance d from the parking lot can be expressed as d 2 4t 4 8 where t represents the time in hours since he left the lake 1 What is an equation for Jada s distance from the parking lot as she heads toward the lake 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 65

G8M4 LESSON 12 2 ZEARN MATH STUDENT EDITION Draw both graphs one representing Han s equation and one representing Jada s equation It is important to be very precise Be careful work in pencil and use a ruler y Distance from parking lot miles 5 4 3 2 1 0 25 66 0 5 0 75 1 1 25 Time hours 1 5 1 75 x 3 Find the point where the two graphs intersect each other What are the coordinates of this point 4 What do the coordinates mean in this situation 5 What has to be true about the relationship between these coordinates and Jada s equation 6 What has to be true about the relationship between these coordinates and Han s equation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 12 ACTIVITY 2 3 A stack of n small cups has a height h in centimeters of h 1 5n 6 A stack of n large cups has a height h in centimeters of h 1 5n 9 Answer the questions about these cups below 1 Graph the equations for each cup on the same set of axes Make sure to label the axes and decide on an appropriate scale 2 For what number of cups will the two stacks have the same height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 67

G8M4 LESSON 12 ZEARN MATH STUDENT EDITION Lesson Summary A system of equations is a set of 2 or more equations where the variables represent the same unknown values For example suppose that two different kinds of bamboo are planted at the same time Plant A starts at 6 ft tall and grows at a constant rate of 14 foot each day Plant B starts at 3 ft tall and grows at a constant rate of 12 foot each day We can write equations y 41 x 6 for Plant A and y 21 x 3 for Plant B where x represents the number of days after being planted and y represents the height We can write this system of equations y 14x 6 y 12x 3 Solving a system of equations means to find the values of x and y that make both equations true at the same time One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point The intersection point represents the pair of x and y values that make both equations true Here is a graph for the bamboo example y 12 11 10 12 9 9 Height ft 8 Plant A 7 6 5 Plant B 4 3 2 1 1 2 3 4 5 6 7 8 Time days 9 10 11 12 13 x The solution to this system of equations is 12 9 which means that both bamboo plants will be 9 feet tall after 12 days We have seen systems of equations that have no solutions one solution and infinitely many solutions When the lines do not intersect there is no solution Lines that do not intersect are parallel When the lines intersect once there is one solution When the lines are right on top of each other there are infinitely many solutions In future lessons we will see that some systems cannot be easily solved by graphing but can be easily solved using algebra TERMINOLOGY System of equations 68 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 12 Name Date GRADE 8 MISSION 4 LESSON 12 Exit Ticket Determined to finish her milkshake before Diego Lin now drinks her 12 ounce milkshake at a rate of 13 an ounce per second Diego starts with his usual 20 ounce milkshake and drinks at the same rate as before 23 an ounce per second 1 Graph this situation on the axes provided y 20 18 16 Ounces 14 12 10 8 6 4 2 3 6 9 12 15 18 21 24 Seconds 2 27 30 33 36 39 42 x What does the graph tell you about the situation and how many solutions there are 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 69

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ZEARN MATH STUDENT EDITION G8M4 LESSON 13 Lesson 13 Solving Systems of Equations Let s solve systems of equations Warm Up 1 Use the lines to decide whether each statement is true or false Be prepared to explain your reasoning using the lines y 20 y x 10 y 2x 4 15 10 2 8 5 25 20 15 10 5 5 5 10 15 20 25 x 10 15 20 1 A solution to 8 x 10 is 2 2 A solution to 2 2x 4 is 8 3 A solution to x 10 2x 4 is 8 4 A solution to x 10 2x 4 is 2 5 There are no values of x and y that make y x 10 and y 2x 4 true at the same time 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 71

G8M4 LESSON 13 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here are three systems of equations graphed on a coordinate plane y A 25 20 15 10 1 5 y B 25 y C 25 25 20 20 20 15 15 15 10 10 10 5 5 5 5 5 10 15 20 25 x 25 20 15 10 5 5 5 10 15 20 25 x 25 20 15 10 5 5 10 10 10 15 15 15 20 20 20 25 25 25 5 10 15 20 25 x Match each figure to one of the systems of equations shown here y 3x 5 a y 2x 20 y 2x 10 b y 4x 1 y 0 5x 12 c y 2x 27 2 72 Find the solution to each system and then check that your solution is reasonable on the graph 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 13 ACTIVITY 2 3 You will each get a page with some systems of equations Use it to solve the problems below 1 Graph each system of equations carefully on the provided coordinate plane 2 Describe what the graph of a system of equations looks like when it has a 1 solution b 0 solutions c infinitely many solutions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 73

G8M4 LESSON 13 ZEARN MATH STUDENT EDITION Lesson Summary Sometimes it is easier to solve a system of equations without having to graph the equations and look for an intersection point In general whenever we are solving a system of equations written as y some stuff y some other stuff we know that we are looking for a pair of values x y that makes both equations true In particular we know that the value for y will be the same in both equations That means that some stuff some other stuff For example look at this system of equations y 2x 6 y 3x 4 Since the y value of the solution is the same in both equations then we know 2x 6 3x 4 We can solve this equation for x 2x 6 3x 4 5x 6 4 5x 10 x 2 add 3x to each side subtract 6 from each side divide each side by 5 But this is only half of what we are looking for we know the value for x but we need the corresponding value for y Since both equations have the same y value we can use either equation to find the y value y 2 2 6 Or y 3 2 4 In both cases we find that y 2 So the solution to the system is 2 2 We can verify this by graphing both equations in the coordinate plane In general a system of linear equations can have No solutions In this case the lines that correspond to each equation never intersect y Exactly one solution The lines that correspond to each equation intersect in exactly one point 3 4 An infinite number of solutions The graphs of the two equations are the same line 1 4 74 2 2 2 3 2 1 1 1 2 x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 13 Name Date GRADE 8 MISSION 4 LESSON 13 Exit Ticket y x 1 Given the lines shown here what are two possible equations for this system of equations 2 How many solutions does this system of equations have Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 75

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ZEARN MATH STUDENT EDITION G8M4 LESSON 14 Lesson 14 Solving More Systems Let s solve systems of equations Warm Up 1 Solve these without writing anything down x 5 y x 7 y 4 y x 3 x 8 y 11 Concept Exploration ACTIVITY 1 2 Use these systems of equations to answer the questions below y 4 a x 5y 6 y 3x 5 e y 4x 30 y 32 x 7 c x 4 y 3x g x 2y 56 y 7 b x 3y 4 y 3x 10 d y 2x 6 1 3x 4y 10 i x 2y y 3x 2 f y 2x 8 y 3x 2 j 2x y 47 x 2y 15 h y 2x x y 10 l x 2y 1 y 2x 5 k 2x 3y 31 Without solving identify 3 systems that you think would be the least difficult to solve and 3 systems that you think would be the most difficult to solve Be prepared to explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 77

G8M4 LESSON 14 2 ZEARN MATH STUDENT EDITION Choose 4 systems from the previous page to solve At least one should be from your least difficult list and one should be from your most difficult list ACTIVITY 2 3 Tyler was looking at this system of equations x y 5 x y 7 He said Just looking at the system I can see it has no solution If you add two numbers that sum can t be equal to two different numbers Do you agree with Tyler 78 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 14 Lesson Summary When we have a system of linear equations where one of the equations is of the form y stuff or x stuff we can solve it algebraically by using a technique called substitution The basic idea is to replace a variable with an expression it is equal to so the expression is like a substitute for the variable For example let s start with the system y 5x 2x y 9 Since we know that y 5x we can substitute 5x for y in the equation 2x y 9 2x 5x 9 and then solve the equation for x x 3 We can find y using either equation Using the first one y 5 3 So 3 15 is the solution to this system We can verify this by looking at the graphs of the equations in the system y 30 y 5x 20 2x y 9 10 10 8 6 4 2 2 4 6 8 10 x 10 3 15 20 30 Sure enough They intersect at 3 15 We didn t know it at the time but we were actually using substitution in the last lesson as well In that lesson we looked at the system y 2x 6 y 3x 4 and we substituted 2x 6 for y into the second equation to get 2x 6 3x 4 Go back and check for yourself 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 79

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ZEARN MATH STUDENT EDITION Name G8M4 LESSON 14 Date GRADE 8 MISSION 4 LESSON 14 Exit Ticket Solve this system of equations y 2x x y 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 81

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ZEARN MATH STUDENT EDITION G8M4 LESSON 15 Lesson 15 Writing Systems of Equations Let s write systems of equations from real world situations Warm Up Match each system of equations with the number of solutions the system has 1 y 3x 4 a y 43 x 1 4 y 4x 5 b y 2x 7 2x 3y 8 c 4x 6y 17 y 5x 15 d y 5 x 3 1 No solutions 2 One solution 3 Infinitely many solutions Concept Exploration ACTIVITY 1 2 For each situation create a system of equations Then without solving interpret what the solution to the system would tell you about the situation 1 Lin s family is out for a bike ride when her dad stops to take a picture of the scenery He tells the rest of the family to keep going and that he ll catch up Lin s dad spends 5 minutes taking the photo and then rides at 0 24 miles per minute until he meets up with the rest of the family further along the bike path Lin and the rest were riding at 0 18 miles per minute 2 Noah is planning a kayaking trip Kayak Rental A charges a base fee of 15 plus 4 50 per hour Kayak Rental B charges a base fee of 12 50 plus 5 per hour 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 83

G8M4 LESSON 15 ZEARN MATH STUDENT EDITION 3 Diego is making a large batch of pastries The recipe calls for 3 strawberries for every apple Diego used 52 fruits all together 4 Flour costs 0 80 per pound and sugar costs 0 50 per pound An order of flour and sugar weighs 15 pounds and costs 9 00 ACTIVITY 2 Use these systems of equations to answer the questions below 3 y 2x 6 a y x 3 y 0 24x e y 0 18x 0 9 y x 4 c y 34 x 9 y 3x g x y 52 y 5x 4 b y 4x 12 2 4y 7x 6 d 4y 7x 5 3 84 y 4 5x 15 f y 5x 12 5 1 Without solving identify 3 systems that you think would be the least difficult for you to solve and 3 systems you think would be the most difficult Be prepared to explain your reasoning 2 Choose 4 systems to solve At least one should be from your least difficult list and one should be from your most difficult list 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M4 LESSON 15 Lesson Summary We have learned how to solve many kinds of systems of equations using algebra that would be difficult to solve by graphing For example look at y 2x 3 x 2y 7 The first equation says that y 2x 3 so wherever we see y we can substitute the expression 2x 3 instead So the second equation becomes x 2 2x 3 7 We can then solve for x x 4x 6 7 5x 6 7 5x 13 x 13 5 distributive property combine like terms add 6 to each side multiply each side by 15 We know that the y value for the solution is the same for either equation so we can use either equation to solve for it Using the first equation we get y 2 13 5 3 substitute x 13 5 into the equation y 26 5 3 26 multiply 2 13 5 to make 5 15 y 26 5 5 rewrite 3 as 15 5 If we substitute x 13 5 into the other equation x 2y 7 we get the same y value So the solution to the 13 11 system is 5 5 There are many kinds of systems of equations that we will learn how to solve in future grades like 2x 3y 6 x 2y 3 Or even y x2 1 y 2x 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 85

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ZEARN MATH STUDENT EDITION Name G8M4 LESSON 15 Date GRADE 8 MISSION 4 LESSON 15 Exit Ticket Solve y 4x 5 x 2y 5 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 87

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ZEARN MATH STUDENT EDITION G8M4 LESSON 16 Lesson 16 Solving Problems with Systems of Equations Let s solve some gnarly problems Warm Up 1 A car is driving towards home at 0 5 miles per minute If the car is 4 miles from home at t 0 which of the following can represent the distance that the car has left to drive 0 5t 4 0 5t 4 0 5t 4 0 5t Concept Exploration ACTIVITY 1 2 1 Solve each problem Explain or show your reasoning Two friends live 7 miles apart One Saturday the two friends set out on their bikes at 8 am and started riding towards each other One rides at 0 2 miles per minute and the other rides at 0 15 miles per minute At what time will the two friends meet 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 89

G8M4 LESSON 16 90 ZEARN MATH STUDENT EDITION 2 Students are selling grapefruits and nuts for a fundraiser The grapefruits cost 1 each and a bag of nuts cost 10 each They sold 100 items and made 307 How many grapefruits did they sell 3 Jada earns 7 per hour mowing her neighbors lawns Andre gets paid 5 per hour for the first hour of babysitting and 8 per hour for any additional hours he babysits What is the number of hours they both can work so that they get paid the same amount 4 Pause here so your teacher can review your work Then invent another problem that is like one of these but with different numbers Solve your problem 5 Create a visual display that includes The new problem you wrote without the solution Enough work space for someone to show a solution 6 Trade your display with another group and solve each other s new problem Make sure that you explain your solution carefully Be prepared to share this solution with the class 7 When the group that got the problem you invented shares their solution check that their answer is correct 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Grade 8 Mission 5 Functions and Volume

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ZEARN MATH STUDENT EDITION G8M5 LESSON 1 Lesson 1 Inputs and Outputs Let s make some rules Warm Up 1 Study the following statements carefully What value can be used in place of x to create true statements Explain your reasoning 12 3 4 because 12 4 3 6 0 x because 6 x 0 Concept Exploration ACTIVITY 1 2 With your partner follow the directions below Keep the rule cards face down Decide who will go first 1 Player 1 picks up a card and silently reads the rule without showing it to Player 2 2 Player 2 chooses an integer and asks Player 1 for the result of applying the rule to that number 3 Player 1 gives the result without saying how they got it 4 Keep going until Player 2 correctly guesses the rule After each round the players switch roles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 93

G8M5 LESSON 1 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 For each input output rule fill in the table with the output that goes with a given input Add two more input output pairs to the table 1 3 4 add 1 then multiply by 4 7 input output 3 4 7 2 35 42 2 3 4 name the digit in the tenths place 7 input output 3 4 7 2 35 42 3 3 4 write 7 7 input output 3 4 7 2 35 42 Pause here until your teacher directs you to the last rule 94 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 1 4 x divide 1 by the input 1 x input output 3 7 7 3 1 0 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 95

G8M5 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary input rule output An input output rule is a rule that takes an allowable input and uses it to determine an output For example the following diagram represents the rule that takes any number as an input then adds 1 multiplies by 4 and gives the resulting number as an output 3 4 add 1 then multiply by 4 7 In some cases not all inputs are allowable and the rule must specify which inputs will work For example this rule is fine when the input is 2 2 divide 6 by 3 more than the input 3 But if the input is 3 we would need to evaluate 6 0 to get the output 3 divide 6 by 3 more than the input So when we say that the rule is divide 6 by 3 more than the input we also have to say that 3 is not allowed as an input 96 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 1 Name Date GRADE 8 MISSION 5 LESSON 1 Exit Ticket Fill in the table for this input output rule 4 divide by 2 and add 1 3 input output 0 2 8 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 97

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ZEARN MATH STUDENT EDITION G8M5 LESSON 2 Lesson 2 Introduction to Functions Let s learn what a function is Warm Up 1 Here are some numbers in a list Answer the questions below 1 3 12 3 2 41 0 5 1 How many different numbers are in the list 2 Make a new list containing the squares of all these numbers 3 How many different numbers are in the new list 4 Explain why the two lists do not have the same number of different numbers 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 99

G8M5 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 100 Say yes or no for each question below If yes draw an input output diagram If no give examples of two different outputs that are possible for the same input 1 A person is 5 5 feet tall Do you know their height in inches 2 A number is 5 Do you know its square 3 The square of a number is 16 Do you know the number 4 A square has a perimeter of 12 cm Do you know its area 5 A rectangle has an area of 16 cm2 Do you know its length 6 You are given a number Do you know the number that is 15 as big 7 You are given a number Do you know its reciprocal 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 2 ACTIVITY 2 3 Here are the questions from the previous activity For the ones you said yes to write a statement like The height a rubber ball bounces to depends on the height it was dropped from or Bounce height is a function of drop height For all of the ones you said no to write a statement like The day of the week does not determine the temperature that day or The temperature that day is not a function of the day of the week 1 A person is 5 5 feet tall Do you know their height in inches 2 A number is 5 Do you know its square 3 The square of a number is 16 Do you know the number 4 A square has a perimeter of 12 cm Do you know its area 5 A rectangle has an area of 16 cm2 Do you know its length 6 You are given a number Do you know the number that is 15 as big 7 You are given a number Do you know its reciprocal 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 101

G8M5 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary Let s say we have an input output rule that for each allowable input gives exactly one output Then we say the output depends on the input or the output is a function of the input For example the area of a square is a function of the side length because you can find the area from the side length by squaring it So when the input is 10 cm the output is 100 cm2 10 find the area of a square given the side length 100 Sometimes we might have two different rules that describe the same function As long as we always get the same single output from the same input the rules describe the same function TERMINOLOGY Function 102 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 2 Date GRADE 8 MISSION 5 LESSON 2 Exit Ticket You are told that you will have to wait for 5 hours in a line with a group of other people Determine if 1 You know the number of minutes you have to wait 2 You know how many people have to wait For each statement if you answer yes draw an input output diagram and write a statement that describes the way one quantity depends on another If you answer no give an example of 2 outputs that are possible for the same input 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 103

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ZEARN MATH STUDENT EDITION G8M5 LESSON 3 Lesson 3 Equations for Functions Let s find outputs from equations Warm Up 1 Fill in the table of input output pairs for the given rule Write an algebraic expression for the rule in the box in the diagram s the side length of a square A the area of the square Input Output 8 2 2 12 14 s 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 105

G8M5 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Record your answers to these questions in the table provided 2 1 Match each of these descriptions with a diagram a the circumference C of a circle with radius r b the distance in miles d that you would travel in t hours if you drive at 60 miles per hour c the output when you triple the input and subtract 4 d the volume of a cube v given its edge length s 2 Write an equation for each description that expresses the output as a function of the input 3 Find the output when the input is 5 for each equation 4 Name the independent and dependent variables of each equation A B s s v 3 C t 60t d r 2 r C D x Description y 3x 4 a b c d Diagram Equation Input 5 Output Independent variable Dependent variable 106 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 3 ACTIVITY 2 3 Jada had some dimes and quarters that had a total value of 12 50 The relationship between the number of dimes d and the number of quarters q can be expressed by the equation 0 1d 0 25q 12 5 1 If Jada has 4 quarters how many dimes does she have 2 If Jada has 10 quarters how many dimes does she have 3 Is the number of dimes a function of the number of quarters If yes write a rule that starts with d that you can use to determine the output d from a given input q If no explain why not 4 If Jada has 25 dimes how many quarters does she have 5 If Jada has 30 dimes how many quarters does she have 6 Is the number of quarters a function of the number of dimes If yes write a rule that starts with q that you can use to determine the output q from a given input d If no explain why not 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 107

G8M5 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary We can sometimes represent functions with equations For example the area A of a circle is a function of the radius r and we can express this with an equation A r2 We can also draw a diagram to represent this function r r2 A In this case we think of the radius r as the input and the area of the circle A as the output For example if the input is a radius of 10 cm then the output is an area of 100 cm2 or about 314 square cm Because this is a function we can find the area A for any given radius r Since it is the input we say that r is the independent variable and as the output A is the dependent variable Sometimes when we have an equation we get to choose which variable is the independent variable For example if we know that 10A 4B 120 then we can think of A as a function of B and write A 0 4B 12 or we can think of B as a function of A and write B 2 5A 30 TERMINOLOGY Dependent variable Independent variable 108 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 3 Date GRADE 8 MISSION 5 LESSON 3 Exit Ticket The value v of your quarters in cents is a function of n the number of quarters you have 1 Draw an input output diagram to represent this function 2 Write an equation that represents this function 3 Find the output when the input is 10 4 Identify the independent and dependent variables 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 109

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ZEARN MATH STUDENT EDITION G8M5 LESSON 4 Lesson 4 Tables Equations and Graphs of Functions Let s connect equations and graphs of functions Warm Up 1 What do you notice What do you wonder time seconds 72 60 48 36 24 12 25 50 75 100 125 150 175 distance from starting line meters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 111

G8M5 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Use the graphs of three functions below to answer the questions A 40 200 30 150 20 100 10 50 1 1 C B 2 3 50 40 30 20 10 1 2 3 20 40 60 80 100 Match each of these equations to one of the graphs a d 60t where d is the distance in miles that you would travel in t hours if you drove at 60 miles per hour b q 50 0 4d where q is the number of quarters and d is the number of dimes in a pile of coins worth 12 50 c 112 A r2 where A is the area in square centimeters of a circle with radius r centimeters 2 Label each of the axes with the independent and dependent variables and the quantities they represent 3 For each function What is the output when the input is 1 What does this tell you about the situation Label the corresponding point on the graph 4 Find two more input output pairs What do they tell you about the situation Label the corresponding points on the graph 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 4 ACTIVITY 2 3 Kiran was running around the track The graph shows the time t he took to run various distances d The table shows his time in seconds after every three meters time seconds 10 9 8 7 6 5 4 3 2 1 3 6 9 12 15 18 21 distance meters 24 27 30 d 0 3 6 9 12 15 18 21 24 27 t 0 1 0 2 0 3 2 3 8 4 6 6 0 6 9 8 09 9 0 a How long did it take Kiran to run 6 meters b How far had he gone after 6 seconds c Estimate when he had run 19 5 meters d Estimate how far he ran in 4 seconds e Is Kiran s time a function of the distance he has run Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 113

G8M5 LESSON 4 43 ZEARN MATH STUDENT EDITION Priya is running once around the track The graph shows her time given how far she is from her starting point time seconds 72 60 48 36 24 12 25 50 75 100 125 150 175 distance from starting line meters a What was her farthest distance from her starting point b Estimate how long it took her to run around the track c Estimate when she was 100 meters from her starting point d Estimate how far she was from the starting line after 60 seconds e Is Priya s time a function of her distance from her starting point Explain how you know 114 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 4 Lesson Summary Here is the graph showing Noah s run time seconds 10 9 8 7 6 5 4 3 2 1 18 6 3 6 9 12 15 18 21 distance meters 24 27 30 The time in seconds since he started running is a function of the distance he has run The point 18 6 on the graph tells you that the time it takes him to run 18 meters is 6 seconds The input is 18 and the output is 6 The graph of a function is all the coordinate pairs input output plotted in the coordinate plane By convention we always put the input first which means that the inputs are represented on the horizontal axis and the outputs on the vertical axis 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 115

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ZEARN MATH STUDENT EDITION G8M5 LESSON 4 Name Date GRADE 8 MISSION 5 LESSON 4 Exit Ticket Here is the graph of a function showing the amount of money remaining on a subway fare card as a function of the number of rides taken 45 dollars on card 40 35 30 P 7 27 5 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 number of rides 1 What is the output of the function when the input is 10 On the graph plot this point and label its coordinates 2 What is the input to the function when the output is 5 On the graph plot this point and label its coordinates 3 What does point P tell you about the situation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 117

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ZEARN MATH STUDENT EDITION G8M5 LESSON 5 Lesson 5 More Graphs of Functions Let s interpret graphs of functions Warm Up 1 Which graph doesn t belong A B C D 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 119

G8M5 LESSON 5 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 The graph shows the temperature between noon and midnight on one day in a certain city y Temperture degrees F 59 58 57 56 55 54 53 52 51 1 120 2 3 4 5 6 7 8 Time hours after noon 9 10 11 x 1 Was it warmer at 3 00 p m or 9 00 p m 2 Approximately when was the temperature highest 3 Find another time that the temperature was the same as it was at 4 00 p m 4 Did the temperature change more between 1 00 p m and 3 00 p m or between 3 00 p m and 5 00 p m 5 Does this graph show that temperature is a function of time or time is a function of temperature 6 When the input for the function is 8 what is the output What does that tell you about the time and temperature 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 5 ACTIVITY 2 3 The graph shows the amount of garbage produced in the US each year between 1991 and 2013 Garbage thousands of tons y 270 000 250 000 230 000 210 000 190 000 1990 1995 2000 Year 2005 2010 2015 x a Did the amount of garbage increase or decrease between 1999 and 2000 b Did the amount of garbage increase or decrease between 2005 and 2009 c Garbage dump by PublicDomainPictures via Pixabay Public Domain Between 1991 and 1995 the amount of garbage increased for three years and then it decreased in the fourth year Describe how the amount of garbage changed in the years between 1995 and 2000 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 121

G8M5 LESSON 5 43 ZEARN MATH STUDENT EDITION The graph shows the percentage of garbage that was recycled between 1991 and 2013 y Percentage recycled 27 24 21 18 15 1990 1995 2000 Year 2005 2010 2015 x a When was it increasing b When was it decreasing c 122 Tell the story of the change in the percentage of garbage recycled in the US over this time period 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 5 Lesson Summary Here is a graph showing the temperature in a town as a function of time after 8 00 p m Temperature degrees F y 60 57 54 51 48 45 1 2 3 4 5 6 7 8 9 Time hours after 8 p m 10 11 x The graph of a function tells us what is happening in the context it represents In this example the temperature starts out at 60 F at 8 00 p m It decreases during the night reaching its lowest point at 8 hours after 8 00 p m or 4 00 a m Then it starts to increase again 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 123

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ZEARN MATH STUDENT EDITION G8M5 LESSON 5 Name Date GRADE 8 MISSION 5 LESSON 5 Exit Ticket Diego runs a 10 kilometer race and keeps track of his speed Speed kilometers per hour 13 12 11 10 9 0 2 4 6 Distance kilometers 8 10 1 What was Diego s speed at the 5 kilometer mark in the race 2 According to the graph where was Diego when he was going the slowest during the race 3 Describe what happened to Diego s speed in the second half of the race from 5 km to 10 km 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 125

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ZEARN MATH STUDENT EDITION G8M5 LESSON 6 Lesson 6 Even More Graphs of Functions Let s draw a graph from a story Warm Up 1 Here are five pictures of a dog taken at equal intervals of time Diego and Lin drew different graphs to represent this situation Diego s graph Lin s graph They both used time as the independent variable What do you think each one used for the dependent variable Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 127

G8M5 LESSON 6 ZEARN MATH STUDENT EDITION Lesson ACTIVITY 1 2 128 For each situation name the independent and dependent variables pick the graph that best fits the situation or sketch the graph if one isn t provided label the axes answer the question which quantity is a function of which Be prepared to explain your reasoning 1 Jada is training for a swimming race The more she practices the less time it takes for her to swim one lap 2 Andre adds some money to a jar in his room each week for 3 weeks and then takes some out in week 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 6 ACTIVITY 2 3 You will receive tools for creating a visual display With your group create a display that shows your response to each question about the story Noah was at home He got on his bike and rode to his friend s house and stayed there for awhile Then he rode home again Then he rode to the park Then he rode home again 1 Create a set of axes and sketch a graph of this story 2 What are the two quantities Label the axes with their names and units of measure For example if this were a story about pouring water into a pitcher one of your labels might say volume liters 3 Which quantity is a function of which Explain your reasoning 4 Based on your graph is his friend s house or the park closer to Noah s home Explain how you know 5 Read the story and all your responses again Does everything make sense If not make changes to your work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 129

G8M5 LESSON 6 ZEARN MATH STUDENT EDITION Lesson Summary distance Here is a graph showing Andre s distance as a function of time time For a graph representing a context it is important to specify the quantities represented on each axis For example if this is showing distance from home then Andre starts at some distance from home maybe at his friend s house moves further away maybe to a park then returns home If instead the graph is showing distance from school the story may be Andre starts out at home moves further away maybe to a friend s house then goes to school What could the story be if the graph is showing distance from a park 130 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 6 Date GRADE 8 MISSION 5 LESSON 6 Exit Ticket Elena starts to walk home from school but has to turn around and go back because she left something in her locker On her way back home the second time she runs into her friend who invites her to the library to do homework with her She stays at the library and then heads home to do her chores Determine Which graph fits Elena s story What the two quantities are Which quantity is a function of which 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 131

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ZEARN MATH STUDENT EDITION G8M5 LESSON 7 Lesson 7 Connecting Representations of Functions Let s connect tables equations graphs and stories of functions Warm Up 1 1 Here are three different ways of representing functions How are they alike How are they different y 2x 2 b 4 3 2 1 2 1 4 3 2 1 1 1 2 3 4 a 2 3 4 3 p 2 1 0 1 2 3 q 4 2 0 2 4 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 133

G8M5 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 The graph shows the temperature between noon and midnight in City A on a certain day The table shows the temperature T in degrees Fahrenheit for h hours after noon in City B Temperature degrees F 2 59 58 57 56 55 54 53 52 51 1 134 2 3 4 5 6 7 8 9 10 11 Time hours after noon h 1 2 3 4 5 6 T 82 78 75 62 58 59 1 Which city was warmer at 4 00 p m 2 Which city had a bigger change in temperature between 1 00 p m and 5 00 p m 3 How much greater was the highest recorded temperature in City B than the highest recorded temperature in City A during this time 4 Compare the outputs of the functions when the input is 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 7 ACTIVITY 2 3 The volume V of a cube with edge length s cm is given by the equation V s3 The volume of a sphere is a function of its radius in centimeters and the graph of this relationship is shown here v 300 250 200 150 100 50 0 5 1 1 5 2 2 5 3 3 5 4 r 1 Is the volume of a cube with edge length s 3 greater or less than the volume of a sphere with radius 3 2 If a sphere has the same volume as a cube with edge length 5 estimate the radius of the sphere 3 Compare the outputs of the two volume functions when the inputs are 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 135

G8M5 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary Functions are all about getting outputs from inputs For each way of representing a function equation graph table or verbal description we can determine the output for a given input Let s say we have a function represented by the equation y 3x 2 where y is the dependent variable and x is the independent variable If we wanted to find the output that goes with 2 we can input 2 into the equation for x and find the corresponding value of y In this case when x is 2 y is 8 since 3 2 2 8 If we had a graph of this function instead then the coordinates of points on the graph are the inputoutput pairs So we would read the y coordinate of the point on the graph that corresponds to a value of 2 for x Looking at the graph of this function here we can see the point 2 8 on it so the output is 8 when the input is 2 y 8 2 8 6 4 2 1 1 x 2 A table representing this function shows the input output pairs directly although only for select inputs x 1 0 1 2 3 y 1 2 5 8 11 Again the table shows that if the input is 2 the output is 8 136 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 7 Name Date GRADE 8 MISSION 5 LESSON 7 Exit Ticket The table shows the area of a square for specific side lengths Side length inches 0 5 1 2 3 Area square inches 0 25 1 4 9 The area A of a circle with radius r is given by the equation A r2 Is the area of a square with side length 2 inches greater than or less than the area of a circle with radius 1 2 inches 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 137

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ZEARN MATH STUDENT EDITION G8M5 LESSON 8 Lesson 8 Linear Functions Let s investigate linear functions Warm Up 1 Diego said that these graphs are ordered from smallest to largest Mai said they are ordered from largest to smallest But these are graphs not numbers What do you think Diego and Mai are thinking Concept Exploration ACTIVITY 1 2 Jada earns 7 per hour mowing her neighbors lawns Answer the questions below 1 Name the two quantities in this situation that are in a functional relationship Which did you choose to be the independent variable What is the variable that depends on it 2 Write an equation that represents the function 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 139

G8M5 LESSON 8 3 Here is a graph of the function Label the axes Label at least two points with input output pairs 3 140 ZEARN MATH STUDENT EDITION Answer the questions about feet and yards below Remember to convert feet to yards you multiply the number of feet by 31 1 Name the two quantities in this situation that are in a functional relationship Which did you choose to be the independent variable What is the variable that depends on it 2 Write an equation that represents the function 3 Draw the graph of the function Label at least two points with input output pairs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 8 ACTIVITY 2 43 There are four tanks of water The amount of water in gallons A in Tank A is given by the function A 200 8t where t is in minutes The amount of water in gallons B in Tank B starts at 400 gallons and is decreasing at 5 gallons per minute These functions work when t 0 and t 80 1 Which tank started out with more water 2 Write an equation representing the relationship between B and t 3 One tank is filling up The other is draining out Which is which How can you tell 4 The amount of water in gallons C in Tank C is given by the function C 800 7t Is it filling up or draining out Can you tell just by looking at the equation 5 The graph of the function for the amount of water in gallons D in Tank D at time t is shown Is it filling up or draining out How do you know D t 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 141

G8M5 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary Suppose a car is traveling at 30 miles per hour The relationship between the time in hours and the distance in miles is a proportional relationship We can represent this relationship with an equation of the form d 30t where distance is a function of time since each input of time has exactly one output of 1 distance Or we could write the equation t 30 d instead where time is a function of distance since each input of distance has exactly one output of time More generally if we represent a linear function with an equation like y mx b then b is the initial value which is 0 for proportional relationships and m is the rate of change of the function If m is positive the function is increasing If m is negative the function is decreasing If we represent a linear function in a different way say with a graph we can use what we know about graphs of lines to find the m and b values and if needed write an equation 142 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 8 Name Date GRADE 8 MISSION 5 LESSON 8 Exit Ticket In a certain city in France they gain 2 minutes of daylight each day after the spring equinox usually in March but after the autumnal equinox usually in September they lose 2 minutes of daylight each day 1 Which of the graphs is most likely to represent the graph of daylight for the month after the spring equinox 2 Which of the graphs is most likely to represent the graph of daylight for the month after the autumnal equinox 3 Why are the other graphs not likely to represent either month Minutes of sunlight B Minutes of sunlight A Days past the equinox Minutes of sunlight D Minutes of sunlight C Days past the equinox Days past the equinox Days past the equinox 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 143

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ZEARN MATH STUDENT EDITION G8M5 LESSON 9 Lesson 9 Linear Models Let s model situations with linear functions Warm Up 1 A candle is burning It starts out 12 inches long After 1 hour it is 10 inches long After 3 hours it is 5 5 inches long 1 When do you think the candle will burn out completely 2 Is the height of the candle a function of time If yes is it a linear function Explain your thinking This graph is here for you to use if you choose y x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 145

G8M5 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 When the sun was directly overhead the stick had no shadow After 20 minutes the shadow was 10 5 cm long After 60 minutes it was 26 cm long 1 Based on this information estimate how long it will be after 95 minutes 2 After 95 minutes the shadow measured 38 5 cm How does this compare to your estimate 3 Is the length of the shadow a function of time If so is it linear Explain your reasoning This graph is here for you to use if you choose y x 146 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 9 ACTIVITY 2 3 Use this graph that shows the percentage of all garbage in the U S that was recycled between 1991 and 2013 to answer the questions below y Percentage recycled 27 24 21 18 15 1990 1995 2000 Year 2005 2010 2015 x 1 Sketch a linear function that models the change in the percentage of garbage that was recycled between 1991 and 1995 For which years is the model good at predicting the percentage of garbage that is produced For which years is it not as good 2 Pick another time period to model with a sketch of a linear function For which years is the model good at making predictions For which years is it not very good 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 147

G8M5 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary Water has different boiling points at different elevations At 0 m above sea level the boiling point is 100 C At 2 500 m above sea level the boiling point is 91 3 C If we assume the boiling point of water is a linear function of elevation we can use these two data points to calculate the slope of the line 100 8 7 m 91 3 2 500 0 2 500 This slope means that for each increase of 2 500 m the boiling point of water decreases by 8 7 C Next we already know the y intercept is 100 C from the first point so a linear equation representing the data is 8 7 y 2 500 x 100 This equation is an example of a mathematical model A mathematical model is a mathematical object like an equation a function or a geometric figure that we use to represent a real life situation Sometimes a situation can be modeled by a linear function We have to use judgment about whether this is a reasonable thing to do based on the information we are given We must also be aware that the model may make imprecise predictions or may only be appropriate for certain ranges of values Testing our model for the boiling point of water it accurately predicts that at an elevation of 1 000 m 8 7 above sea level when x 1 000 water will boil at 96 5 C since y 2 500 1 000 100 96 5 For higher elevations the model is not as accurate but it is still close At 5 000 m above sea level it predicts 82 6 C which is 0 6 C off the actual value of 83 2 C At 9 000 m above sea level it predicts 68 7 C which is about 3 C less than the actual value of 71 5 C The model continues to be less accurate at even higher elevations since the relationship between the boiling point of water and elevation isn t linear but for the elevations in which most people live it s pretty good 148 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 9 Date GRADE 8 MISSION 5 LESSON 9 Exit Ticket A small company is selling a new board game and they need to know how many to produce in the future After 12 months they sold 4 thousand games after 18 months they sold 7 thousand games and after 36 months they sold 15 thousand games 1 Could this information be reasonably estimated using a single linear model 2 If so use the model to estimate the number of games sold after 48 months If not explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 149

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ZEARN MATH STUDENT EDITION G8M5 LESSON 10 Lesson 10 Piecewise Linear Functions Let s explore functions built out of linear pieces Warm Up 1 What do you notice What do you wonder y Temperature degrees F 60 59 58 57 56 55 54 53 52 51 50 0 1 2 3 4 5 6 7 8 Time hours after midnight 9 10 11 12 x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 151

G8M5 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Use the graph to answer the questions below y Percentage recycled 27 24 21 18 15 1990 152 1995 2000 Year 2005 2010 2015 x 1 Approximate the percentage recycled each year with a piecewise linear function by drawing between three and five line segments to approximate the graph 2 Find the slope for each piece What do these slopes tell you 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 10 ACTIVITY 2 3 Elena filled up the tub and gave her dog a bath Then she let the water out of the tub Use the graph to answer the questions below y 30 27 24 21 18 15 12 9 6 3 3 6 9 12 15 18 21 24 27 30 x 1 The graph shows the amount of water in the tub in gallons as a function of time in minutes Add labels to the graph to show this 2 When did she turn off the water faucet 3 How much water was in the tub when she bathed her dog 4 How long did it take for the tub to drain completely 5 At what rate did the faucet fill the tub 6 At what rate did the water drain from the tub 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 153

G8M5 LESSON 10 ZEARN MATH STUDENT EDITION Lesson Summary This graph shows Andre biking to his friend s house where he hangs out for a while Then they bike together to the store to buy some groceries before racing back to Andre s house for a movie night Each line segment in the graph represents a different part of Andre s travels Distance from home y x Time This is an example of a piecewise linear function which is a function whose graph is pieced together out of line segments It can be used to model situations in which a quantity changes at a constant rate for a while then switches to a different constant rate We can use piecewise functions to represent stories or we can use them to model actual data In the second example temperature recordings at several times throughout a day are modeled with a piecewise function made up of two line segments Which line segment do you think does the best job of modeling the data y Temperature degrees F 60 59 58 57 56 55 54 53 52 51 50 0 154 1 2 3 4 5 6 7 8 Time hours after midnight 9 10 11 12 x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 10 Name Date GRADE 8 MISSION 5 LESSON 10 Exit Ticket Lin uses an app to graph the charge on her phone y 100 Percentage change 80 60 40 20 1 2 3 4 5 6 7 8 9 10 11 x Hours after noon 1 When did she start using her phone 2 When did she start charging her phone 3 While she was using her phone at what rate was Lin s phone battery dying 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 155

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ZEARN MATH STUDENT EDITION G8M5 LESSON 11 Lesson 11 Filling Containers Let s fill containers with water Warm Up 1 These are drawings of three dimensional objects Which one doesn t belong Explain your reasoning A B C D 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 157

G8M5 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 You will receive a graduated cylinder water and some other supplies Your group will use these supplies to investigate the height of water in the cylinder as a function of the water volume 1 Before you get started make a prediction about the shape of the graph 2 Fill the cylinder with different amounts of water and record the data in the table volume ml height cm 158 3 Create a graph that shows the height of the water in the cylinder as a function of the water volume 4 Choose a point on the graph and explain its meaning in the context of the situation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 11 ACTIVITY 2 3 The graph shows the height vs volume function of an unknown container What shape could this container have Explain how you know and draw a possible container height in centimeters 1 Use the graphs below to answer the questions volume in milliliters 2 The graph shows the height vs volume function of a different unknown container What shape could this container have Explain how you know and draw a possible container height in centimeters 14 12 10 8 6 4 2 10 3 20 30 40 50 60 70 volume in milliliters 80 90 100 How are the two containers similar How are they different 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 159

G8M5 LESSON 11 ZEARN MATH STUDENT EDITION Lesson Summary When filling a shape like a cylinder with water we can see how the dimensions of the cylinder affect things like the changing height of the water For example let s say we have two cylinders D and E with the same height but D has a radius of 3 cm and E has a radius of 6 cm D E h 3 cm h 6 cm If we pour water into both cylinders at the same rate the height of water in D will increase faster than the height of water in E due to its smaller radius This means that if we made graphs of the height of water as a function of the volume of water for each cylinder we would have two lines and the slope of the line for cylinder D would be greater than the slope of the line for cylinder E 160 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 11 Date GRADE 8 MISSION 5 LESSON 11 Two cylinders a and b each started with different amounts of water The graph shows how the height of the water changed as the volume of water increased in each cylinder Which cylinder has the larger radius Explain how you know height in centimeters Exit Ticket a b volume in milliliters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 161

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ZEARN MATH STUDENT EDITION G8M5 LESSON 12 Lesson 12 How Much Will Fit Let s reason about the volume of different shapes Warm Up 1 The small container holds 200 beans Estimate how many beans the large container holds Lesson ACTIVITY 1 2 You will see some different containers Use these containers to answer the questions below 1 If the pasta box holds 8 cups of rice how much rice would you need for the other rectangular prisms 2 If the pumpkin can holds 15 fluid ounces of rice how much do the other cylinders hold 3 If the small cone holds 2 fluid ounces of rice how much does the large cone hold 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 163

G8M5 LESSON 12 4 ZEARN MATH STUDENT EDITION If the golf ball were hollow it would hold about 0 2 cups of water If the softball were hollow how much would the sphere hold ACTIVITY 2 3 A 164 Use the images below to answer the questions B C D 1 What shapes are the faces of each type of object shown here For example all six faces of a cube are squares 2 Which faces could be referred to as a base of the object 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 43 G8M5 LESSON 12 Practice sketching some cylinders Sketch a few different sizes including short tall narrow wide and sideways Label the radius r and height h on each cylinder Here is a method for quickly sketching a cylinder Draw two ovals Connect the edges Which parts of your drawing would be hidden behind the cylinder Make these parts dashed lines 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 165

G8M5 LESSON 12 ZEARN MATH STUDENT EDITION Lesson Summary The volume of a three dimensional figure like a jar or a room is the amount of space the shape encloses We can measure volume by finding the number of equal sized volume units that fill the figure without gaps or overlaps For example we might say that a room has a volume of 1 000 cubic feet or that a pitcher can carry 5 gallons of water We could even measure volume of a jar by the number of beans it could hold though a bean count is not really a measure of the volume in the same way that a cubic centimeter is because there is space between the beans The number of beans that fit in the jar do depend on the volume of the jar so it is an okay estimate when judging the relative sizes of containers In earlier grades we studied three dimensional figures with flat faces that are polygons We learned how to calculate the volumes of rectangular prisms Now we will study three dimensional figures with circular faces and curved surfaces cones cylinders and spheres To help us see the shapes better we can use dotted lines to represent parts that we wouldn t be able to see if a solid physical object were in front of us For example if we think of the cylinder in this picture as representing a tin can the dotted arc in the bottom half of that cylinder represents the back half of the circular base of the can What objects could the other figures in the picture represent 166 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 12 Date GRADE 8 MISSION 5 LESSON 12 Exit Ticket Here is a box of pasta and a cylindrical container The two objects are the same height and the cylinder is just wide enough for the box to fit inside with all 4 vertical edges of the box touching the inside of the cylinder If the box of pasta fits 8 cups of rice estimate how many cups of rice will fit inside the cylinder Explain or show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 167

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ZEARN MATH STUDENT EDITION G8M5 LESSON 13 Lesson 13 The Volume of a Cylinder Let s explore cylinders and their volumes Warm Up 1 Here is a circle Points A B C and D are drawn as well as Segments AD and BC Answer the questions below D 4 B A C 1 What is the area of the circle in square units Select all that apply a 4 b 8 c 16 d 42 e approximately 25 f 2 approximately 50 If the area of a circle is 49 square units what is its radius Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 169

G8M5 LESSON 13 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 What is the volume of each figure in cubic units Even if you aren t sure make a reasonable guess 8 2 4 4 A 170 B 1 C 1 Figure A A rectangular prism whose base has an area of 16 square units and whose height is 3 units 2 Figure B A cylinder whose base has an area of 16 square units and whose height is 1 unit 3 Figure C A cylinder whose base has an area of 16 square units and whose height is 3 units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 13 ACTIVITY 2 3 Here is a cylinder with height 4 units and diameter 10 units 4 10 a Shade the cylinder s base b What is the area of the cylinder s base Express your answer in terms of c 43 What is the volume of this cylinder Express your answer in terms of A silo is a cylindrical container that is used on farms to hold large amounts of goods such as grain On a particular farm a silo has a height of 18 feet and diameter of 6 feet Make a sketch of this silo and label its height and radius How many cubic feet of grain can this silo hold Use 3 14 as an approximation for 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 171

G8M5 LESSON 13 ZEARN MATH STUDENT EDITION Lesson Summary We can find the volume of a cylinder with radius r and height h using two ideas we ve seen before The volume of a rectangular prism is a result of multiplying the area of its base by its height The base of the cylinder is a circle with radius r so the base area is r2 Remember that is the number we get when we divide the circumference of any circle by its diameter The value of is approximately 3 14 Just like a rectangular prism the volume of a cylinder is the area of the base times the height For example take a cylinder whose radius is 2 cm and whose height is 5 cm 5 2 The base has an area of 4 cm2 since 22 4 so the volume is 20 cm3 since 4 5 20 Using 3 14 as an approximation for we can say that the volume of the cylinder is approximately 62 8 cm3 In general the base of a cylinder with radius r units has area r2 square units If the height is h units then the volume V in cubic units is V r2h 172 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 13 Date GRADE 8 MISSION 5 LESSON 13 Exit Ticket The cylinder shown here has a height of 7 centimeters and a radius of 4 centimeters 1 What is the area of the base of the cylinder Express your answer in terms of 2 How many cubic centimeters of fluid can fill this cylinder Express your answer in terms of 3 Give a decimal approximation of your answer to the second question using 3 14 to approximate 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 173

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ZEARN MATH STUDENT EDITION G8M5 LESSON 14 Lesson 14 Finding Cylinder Dimensions Let s figure out the dimensions of cylinders Warm Up 1 What is a possible volume for this cylinder if the diameter is 8 cm Explain your reasoning h 8 Concept Exploration ACTIVITY 1 2 The volume V of a cylinder with radius r is given by the formula V r2h The volume of this cylinder with radius 5 units is 50 cubic units This statement is true 50 52 h h 5 What does the height of this cylinder have to be Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 175

G8M5 LESSON 14 3 ZEARN MATH STUDENT EDITION The volume V of a cylinder with radius r is given by the formula V r2h The volume of this cylinder with height 4 units is 36 cubic units This statement is true 36 r2 4 4 r What does the radius of this cylinder have to be Explain how you know ACTIVITY 2 43 Each row of the table has information about a particular cylinder Complete the table with the missing dimensions d r h 176 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Diameter units G8M5 LESSON 14 Radius units 3 Area of the base square units Height units Volume cubic units 5 12 108 11 8 99 16 100 10 16 20 20 314 b b a2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 177

G8M5 LESSON 14 ZEARN MATH STUDENT EDITION Lesson Summary In an earlier lesson we learned that the volume V of a cylinder with radius r and height h is V r2h We say that the volume depends on the radius and height and if we know the radius and height we can find the volume It is also true that if we know the volume and one dimension either radius or height we can find the other dimension For example imagine a cylinder that has a volume of 550 cm3 and a radius of 5 cm but the height is unknown From the volume formula we know that 500 25 h must be true Looking at the structure of the equation we can see that 500 25h That means that the height has to be 20 cm since 500 25 20 Now imagine another cylinder that also has a volume of 500 cm3 with an unknown radius and a height of 5 cm Then we know that 500 r2 5 must be true Looking at the structure of this equation we can see that r2 100 So the radius must be 10 cm 178 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 14 Name Date GRADE 8 MISSION 5 LESSON 14 Exit Ticket h r This cylinder has a volume of 12 cubic inches and a diameter of 4 inches Find the cylinder s radius and height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 179

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ZEARN MATH STUDENT EDITION G8M5 LESSON 15 Lesson 15 The Volume of a Cone Let s explore cones and their volumes Warm Up 1 The cone and cylinder have the same height and the radii of their bases are equal 1 Which figure has a larger volume 2 Do you think the volume of the smaller one is more or less than 12 the volume of the larger one Explain your reasoning 3 Here is a method for quickly sketching a cone 8 8 Draw an oval Draw a point centered above the oval Connect the edges of the oval to the point Which parts of your drawing would be hidden behind the object Make these parts dashed lines Sketch two different sized cones The oval doesn t have to be on the bottom For each drawing label the cone s radius with r and height with h 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 181

G8M5 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 182 A cone and cylinder have the same height and their bases are congruent circles 1 If the volume of the cylinder is 90 cm3 what is the volume of the cone 2 If the volume of the cone is 120 cm3 what is the volume of the cylinder 3 If the volume of the cylinder is V r2h what is the volume of the cone Either write an expression for the cone or explain the relationship in words 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 15 ACTIVITY 2 3 1 2 Use the figures to solve the problems below Here is a cylinder and cone that have the same height and the same base area What is the volume of each figure Express your answers in terms of 4 10 Here is a cone a What is the area of the base Express your answer in terms of 8 6 b What is the volume of the cone Express your answer in terms of 43 A cone shaped popcorn cup has a radius of 5 centimeters and a height of 9 centimeters How many cubic centimeters of popcorn can the cup hold Use 3 14 as an approximation for and give a numerical answer 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 183

G8M5 LESSON 15 ZEARN MATH STUDENT EDITION Lesson Summary If a cone and a cylinder have the same base and the same height then the volume of the cone is 13 of the volume of the cylinder For example the cylinder and cone shown here both have a base with radius 3 feet and a height of 7 feet The cylinder has a volume of 63 cubic feet since 32 7 63 The cone has a volume that is 13 of that or 21 cubic feet 7 7 3 3 If the radius for both is r and the height for both is h then the volume of the cylinder is r2h That means that the volume V of the cone is V 13 r2h 184 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 15 Date GRADE 8 MISSION 5 LESSON 15 Exit Ticket A cone with the same base but a height 3 times taller than the given cylinder exists What is the volume of each figure Express your answers in terms of 4 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 185

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ZEARN MATH STUDENT EDITION G8M5 LESSON 16 Lesson 16 Finding Cone Dimensions Let s figure out the dimensions of cones Warm Up 1 For each equation decide what value if any would make it true 1 27 13 h 2 27 13 r2 3 12 13 a 4 12 13 b2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 187

G8M5 LESSON 16 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Each row of the table has some information about a particular cone Complete the table with the missing dimensions diameter units radius units area of the base square units height units 4 3 1 3 6 36 1 4 20 188 volume of cone cubic units 200 12 64 3 3 14 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 16 ACTIVITY 2 3 A movie theater offers two containers 19 cm 12 cm 6 75 15 cm 8 cm 6 25 Which container is the better value Use 3 14 as an approximation for 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 189

G8M5 LESSON 16 ZEARN MATH STUDENT EDITION Lesson Summary As we saw with cylinders the volume V of a cone depends on the radius r of the base and the height h V 13 r2 h If we know the radius and height we can find the volume If we know the volume and one of the dimensions either radius or height we can find the other dimension For example imagine a cone with a volume of 64 cm3 a height of 3 cm and an unknown radius r From the volume formula we know that 64 13 r2 3 Looking at the structure of the equation we can see that r2 64 so the radius must be 8 cm Now imagine a different cone with a volume of 18 cm3 a radius of 3 cm and an unknown height h Using the formula for the volume of the cone we know that 18 13 32 h so the height must be 6 cm Can you see why 190 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 16 Date GRADE 8 MISSION 5 LESSON 16 Exit Ticket Noah and Lin are making paper cones to hold popcorn to hand out at parent math night They want the cones to hold 9 cubic inches of popcorn What are two different possible values for height h and radius r for the cones 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 191

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ZEARN MATH STUDENT EDITION G8M5 LESSON 17 Lesson 17 Scaling One Dimension Let s see how changing one dimension changes the volume of a shape Warm Up 1 2 Here is a graph of the amount of gas burned during a trip by a tractor trailer truck as it drives at a constant speed down a highway Answer the questions below At the end of the trip how far did the truck drive and how much gas did it use If a truck traveled half this distance at the same rate how much gas would it use 100 90 80 70 60 50 40 30 20 10 gas burned gallons 1 40 80 120 160 200 240 distance traveled miles 3 If a truck traveled double this distance at the same rate how much gas would it use 4 Complete the sentence is a function of 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 193

G8M5 LESSON 17 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 194 There are many right rectangular prisms with one side of length 5 units and another side of length 3 units Let s represent the length of the third side and V represent the volume of these prisms Answer the questions below 1 Write an equation that represents the relationship between V and s 2 Graph this equation and label the axes 3 What happens to the volume if you double the edge length s Where do you see this in the graph Where do you see it algebraically 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 17 ACTIVITY 2 3 There are many cylinders with radius 5 units Let h represent the height and V represent the volume of these cylinders Answer the questions below 1 Write an equation that represents the relationship between V and h Use 3 14 as an approximation of 2 Graph this equation and label the axes 3 What happens to the volume if you halve the height h Where can you see this in the graph How can you see it algebraically 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 195

G8M5 LESSON 17 ZEARN MATH STUDENT EDITION ACTIVITY 3 Here is a graph of the relationship between the height and the volume of some cones that all have the same radius Answer the questions below volume of cone 43 4000 3000 10 2355 2000 1000 1 2 3 4 5 6 7 8 9 10 11 12 height of cone 196 1 What do the coordinates of the labeled point represent 2 What is the volume of the cone with height 5 With height 30 3 Use the labeled point to find the radius of these cones Use 3 14 as an approximation for 4 Write an equation that relates the volume V and height h 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 17 Lesson Summary Imagine a cylinder with a radius of 5 cm that is being filled with water As the height of the water increases the volume of water increases We say that the volume of the water in the cylinder V depends on the height of the water h We can represent this relationship with an equation V 52h or just V 25 h This equation represents a proportional relationship between the height and the volume We can use this equation to understand how the volume changes when the height is tripled 3h h 5 5 The new volume would be V 25 3h 75 h which is precisely 3 times as much as the old volume of 25 h In general when one quantity in a proportional relationship changes by a given factor the other quantity changes by the same factor Remember that proportional relationships are examples of linear relationships which can also be thought of as functions So in this example V the volume of water in the cylinder is a function of the height h of the water 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 197

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ZEARN MATH STUDENT EDITION G8M5 LESSON 17 Name Date GRADE 8 MISSION 5 LESSON 17 volume ft3 Exit Ticket 65 60 55 50 45 50 35 30 25 20 15 10 5 18 56 52 9 28 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 height ft Here is a graph of the relationship between the height and volume of some cylinders that all have the same radius R An equation that represents this relationship is V R2h use 3 14 as an approximation for What is the radius of these cylinders 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 199

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ZEARN MATH STUDENT EDITION G8M5 LESSON 18 Lesson 18 Scaling Two Dimensions Let s change more dimensions of shapes Warm Up 1 m n a b and c all represent positive integers Consider these two equations m a b c n abc 1 Which of these statements are true Select all that apply a If a is tripled m is tripled b If a b and c are all tripled then m is tripled c If a is tripled n is tripled d If a b and c are all tripled then n is tripled 2 Create a true statement of your own about one of the equations Concept Exploration ACTIVITY 1 2 Clare sketches a rectangular prism with a height of 11 and a square base and labels the edges of the base s She asks Han what he thinks will happen to the volume of the rectangular prism if she triples s Han says the volume will be 9 times bigger Is he right Explain or show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 201

G8M5 LESSON 18 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 202 There are many cones with a height of 7 units Let r represent the radius and V represent the volume of these cones 1 Write an equation that expresses the relationship between V and r Use 3 14 as an approximation for 2 Predict what happens to the volume if you triple the value of r 3 Graph this equation 4 What happens to the volume if you triple r Where do you see this in the graph How can you see it algebraically 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 18 Lesson Summary There are many rectangular prisms that have a length of 4 units and width of 5 units but differing heights If h represents the height then the volume V of such a prism is V 20h The equation shows us that the volume of a prism with a base area of 20 square units is a linear function of the height Because this is a proportional relationship if the height gets multiplied by a factor of a then the volume is also multiplied by a factor of a V 20 ah What happens if we scale two dimensions of a prism by a factor of a In this case the volume gets multiplied by a factor of a twice or a2 For example think about a prism with a length of 4 units width of 5 units and height of 6 units Its volume is 120 cubic units since 4 5 6 120 Now imagine the length and width each get scaled by a factor of a meaning the new prism has a length of 4a width of 5a and a height of 6 The new volume is 120a2 cubic units since 4a 5a 6 120a2 A similar relationship holds for cylinders Think of a cylinder with a height of 6 and a radius of 5 The volume would be 150 cubic units since 52 6 150 Now imagine the radius is scaled by a factor of a Then the new volume is 5a 2 6 25a2 6 or 150a2 cubic units So scaling the radius by a factor of a has the effect of multiplying the volume by a2 Why does the volume multiply by a2 when only the radius changes This makes sense if we imagine how scaling the radius changes the base area of the cylinder As the radius increases the base area gets larger in two dimensions the circle gets wider and also taller while the third dimension of the cylinder height stays the same 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 203

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ZEARN MATH STUDENT EDITION Name G8M5 LESSON 18 Date GRADE 8 MISSION 5 LESSON 18 Exit Ticket There are many cylinders for which the height and radius are the same value Let c represent the height and radius of a cylinder and V represent the volume of the cylinder 1 Write an equation that expresses the relationship between the volume height and radius of this cylinder using c and V 2 If the value of c is halved what must happen to the value of the volume V 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 205

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ZEARN MATH STUDENT EDITION G8M5 LESSON 19 Lesson 19 Estimating a Hemisphere Let s estimate volume of hemispheres with figures we know Warm Up 1 Here are two shapes What do you notice What do you wonder V 1 3 r3 V r3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 207

G8M5 LESSON 19 ZEARN MATH STUDENT EDITION Lesson ACTIVITY 1 2 Mai has a dome paperweight that she can use as a magnifier The paperweight is shaped like a hemisphere made of solid glass so she wants to design a box to keep it in so it won t get broken Her paperweight has a radius of 3 cm r r 208 1 What should the dimensions of the inside of box be so the box is as small as possible 2 What is the volume of the box 3 What is a reasonable estimate for the volume of the paperweight 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 G8M5 LESSON 19 Tyler has a different box with side lengths that are twice as long as the sides of Mai s box Tyler s box is just large enough to hold a different glass paperweight 1 What is the volume of the new box 2 What is a reasonable estimate for the volume of this glass paperweight 3 How many times bigger do you think the volume of the paperweight in this box is than the volume of Mai s paperweight Explain your thinking ACTIVITY 2 43 A hemisphere with radius 5 units fits snugly into a cylinder of the same radius and height 5 5 1 Calculate the volume of the cylinder 2 Estimate the volume of the hemisphere Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 209

G8M5 LESSON 19 53 ZEARN MATH STUDENT EDITION A cone fits snugly inside a hemisphere and they share a radius of 5 5 5 1 What is the volume of the cone 2 Estimate the volume of the hemisphere Explain your reasoning 63 210 Compare your estimate for the hemisphere with the cone inside to your estimate of the hemisphere inside the cylinder How do they compare to the volumes of the cylinder and the cone 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 19 Lesson Summary We can estimate the volume of a hemisphere by comparing it to other shapes for which we know the volume For example a hemisphere of radius 1 unit fits inside a cylinder with a radius of 1 unit and height of 1 unit 1 Since the hemisphere is inside the cylinder it must have a smaller volume than the cylinder making the cylinder s volume a reasonable over estimate for the volume of the hemisphere 1 The volume of this particular cylinder is about 3 14 units3 since 1 2 1 so we know the volume of the hemisphere is less than 3 14 cubic units Using similar logic a cone of radius 1 unit and height 1 unit fits inside of the hemisphere of radius 1 unit Since the cone is inside the hemisphere the cone must have a smaller volume than the hemisphere making the cone s volume a reasonable under estimate for the volume of the hemisphere The volume of this particular cone is about 1 05 units3 since 13 1 2 1 13 1 05 so we know the volume of the hemisphere is more than 1 05 cubic units 1 1 Averaging the volumes of the cylinder and the cone we can estimate the volume of the hemisphere to be about 2 10 units3 since 3 14 2 1 05 2 10 And since a hemisphere is half of a sphere we can also estimate that a sphere with radius of 1 would be double this volume or about 4 20 units3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 211

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ZEARN MATH STUDENT EDITION Name G8M5 LESSON 19 Date GRADE 8 MISSION 5 LESSON 19 Exit Ticket A hemisphere shaped security mirror fits exactly inside a rectangular prism box with a square base that has edge length 10 inches What is a reasonable estimate for the volume of this mirror 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 213

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ZEARN MATH STUDENT EDITION G8M5 LESSON 20 Lesson 20 The Volume of a Sphere Let s explore spheres and their volumes Warm Up 1 Use the method below for quickly sketching a sphere to solve the problems Draw a circle Draw an oval in the middle whose edges touch the sphere 1 Practice sketching some spheres Sketch a few different sizes 2 For each sketch draw a radius and label it r 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 215

G8M5 LESSON 20 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 216 Here are a cone a sphere and a cylinder that all have the same radii and heights The radius of the cylinder is 5 units When necessary express all answers in terms of 1 What is the height of the cylinder 2 What is the volume of the cylinder 3 What is the volume of the cone 4 What is the volume of the sphere Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 20 ACTIVITY 2 3 Here are a cone a sphere and a cylinder that all have the same radii and heights Let the radius of the cylinder be r units When necessary express answers in terms of 1 What is the height of the cylinder in terms of r 2 What is the volume of the cylinder in terms of r 3 What is the volume of the cone in terms of r 4 What is the volume of the sphere in terms of r 5 The volume of the cone is 13 the volume of the cylinder The volume of the sphere is what fraction of the volume of the cylinder 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 217

G8M5 LESSON 20 ZEARN MATH STUDENT EDITION Lesson Summary Think about a sphere with radius r units that fits snugly inside a cylinder The cylinder must then also have a radius of r units and a height of 2r units Using what we have learned about volume the cylinder has a volume of r2h r2 2r which is equal to 2 r3 cubic units We know from an earlier lesson that the volume of a cone with the same base and height as a cylinder has 13 of the volume In this example such a cone has a volume of 13 r2 2r or just 23 r3 cubic units 2r r r 2r r If we filled the cone and sphere with water and then poured that water into the cylinder the cylinder would be completely filled That means the volume of the sphere and the volume of the cone add up to the volume of the cylinder In other words if V is the volume of the sphere then V 32 r3 2 r3 This leads to the formula for the volume of the sphere V 43 r3 218 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 20 Date GRADE 8 MISSION 5 LESSON 20 Exit Ticket Recall that the volume of a sphere is given by the formula V 34 r3 4 1 Here is a sphere with radius 4 feet What is the volume of the sphere Express your answer in terms of 2 A spherical balloon has a diameter of 4 feet Approximate how many cubic feet of air this balloon holds Use 3 14 as an approximation for and give a numerical answer 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 219

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ZEARN MATH STUDENT EDITION G8M5 LESSON 21 Lesson 21 Cylinders Cones and Spheres Let s find the volumes of shapes Warm Up 1 Four students each calculated the volume of a sphere with a radius of 9 centimeters and they got four different answers Han thinks it is 108 cubic centimeters Jada got 108 cubic centimeters Tyler calculated 972 cubic centimeters Mai says it is 972 cubic centimeters Do you agree with any of them Explain your reasoning Concept Exploration ACTIVITY 1 2 The volume of this sphere with radius r is V 288 r This statement is true 288 43 r3 What is the value of r for this sphere Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 221

G8M5 LESSON 21 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 You will each get either a problem card or a data card Do not show or read your card to your partner If your teacher gives you the problem card If your teacher gives you the data card 1 1 Silently read the information on your card 2 Ask your partner What specific information do you need and wait for your partner to ask for information Only give information that is on your card Do not figure out anything for your partner Silently read your card and think about what information you need to answer the question 2 Ask your partner for the specific information that you need 3 Explain to your partner how you are using the information to solve the problem 3 Solve the problem and explain your reasoning to your partner Before telling your partner the information ask Why do you need that information 4 After your partner solves the problem ask them to explain their reasoning and listen to their explanation 4 Pause here so your teacher can review your work Ask your teacher for a new set of cards and repeat the activity trading roles with your partner 222 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 21 ACTIVITY 3 43 1 A cylinder with a diameter of 3 centimeters and a height of 8 centimeters is filled with water Decide which figures described here if any could hold all of the water from the cylinder Explain your reasoning Cone with a height of 8 centimeters and a radius of 3 centimeters 8 cm 3 cm 2 Cylinder with a diameter of 6 centimeters and a height of 2 centimeters 3 Rectangular prism with a length of 3 centimeters width of 4 centimeters and height of 8 centimeters 4 Sphere with a radius of 2 centimeters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 223

G8M5 LESSON 21 ZEARN MATH STUDENT EDITION Lesson Summary The formula V 43 r3 gives the volume of a sphere with radius r We can use the formula to find the volume of a sphere with a known radius For example if the radius of a sphere is 6 units then the volume would be 4 3 3 6 288 or approximately 904 cubic units We can also use the formula to find the radius of a sphere if we only know its volume For example if we know the volume of a sphere is 36 cubic units but we don t know the radius then this equation is true 36 43 r3 That means that r3 27 so the radius r has to be 3 units in order for both sides of the equation to have the same value Many common objects from water bottles to buildings to balloons are similar in shape to rectangular prisms cylinders cones and spheres or even combinations of these shapes Using the volume formulas for these shapes allows us to compare the volumes of different types of objects sometimes with surprising results For example a cube shaped box with a side length of 3 centimeters holds less than a sphere with a radius of 2 centimeters because the volume of the cube is 27 cubic centimeters 33 27 and the volume of the sphere is around 33 51 cubic centimeters 43 23 33 51 224 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M5 LESSON 21 Date GRADE 8 MISSION 5 LESSON 21 Exit Ticket Some information is given about each sphere Order them from least volume to greatest volume You may sketch a sphere to help you visualize if you prefer Sphere A Has a radius of 4 Sphere B Has a diameter of 6 Sphere C Has a volume of 64 Sphere D Has a radius double that of sphere B 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 225

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ZEARN MATH STUDENT EDITION G8M5 LESSON 22 Lesson 22 Volume As a Function of Let s compare water heights in different containers Warm Up 1 A cylinder and sphere have the same height 1 If the sphere has a volume of 36 cubic units what is the height of the cylinder 2 What is a possible volume for the cylinder Be prepared to explain your reasoning Lesson ACTIVITY 1 2 Fill in the missing volumes in terms of Add two more radius and volume pairs of your choosing Then answer the questions below 1 Radius 1 Volume 4 3 2 3 1 2 1 3 100 r a How does the volume of a sphere with radius 2 cm compare to the volume of a sphere with radius 1 cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 227

G8M5 LESSON 22 ZEARN MATH STUDENT EDITION b How does the volume of a sphere with radius 21 cm compare to the volume of a sphere with radius 1 cm 2 A sphere has a radius of length r a What happens to the volume of this sphere if its radius is doubled b What happens to the volume of this sphere if its radius is halved 3 228 Sphere Q has a volume of 500 cm3 Sphere S has a radius 15 as large as Sphere Q What is the volume of Sphere S 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M5 LESSON 22 ACTIVITY 2 43 Three containers of the same height were filled with water at the same rate One container is a cylinder one is a cone and one is a sphere As they were filled the relationship between the volume of water and the height of the water was recorded in different ways shown below Cone Height in V Cylinder h 4 Sphere y Volume in3 Height in 7 0 0 6 8 38 1 5 29 32 2 4 56 55 3 3 83 76 4 2 104 72 5 1 113 04 6 120 6 200 6 20 40 60 80 Volume in3 100 120 x 1 The maximum volume of water the cylinder can hold is 24 What is the radius of the cylinder 2 Graph the relationship between the volume of water poured into the cylinder and the height of water in the cylinder on the same axes as the cone What does the slope of this line represent 3 Which container can fit the largest volume of water The smallest 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 229

G8M5 LESSON 22 230 ZEARN MATH STUDENT EDITION 4 About how much water does it take for the cylinder and the sphere to have the same height The cylinder and the cone Explain how you know 5 For what approximate range of volumes is the height of the water in the cylinder greater than the height of the water in the cone Explain how you know 6 For what approximate range of volumes is the height of the water in the sphere less than the height of the water in the cylinder Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Grade 8 Mission 6 Associations in Data

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ZEARN MATH STUDENT EDITION G8M6 LESSON 1 Lesson 1 Organizing Data Let s find ways to show patterns in data Warm Up 1 Here is a table of data Each row shows two measurements of a triangle What do you notice What do you wonder Length of short side cm Length of perimeter cm 0 25 1 2 7 5 6 5 22 3 9 5 0 5 2 1 25 4 5 3 5 12 5 1 5 5 4 14 1 3 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 233

G8M6 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Here is the table of isosceles right triangle measurements from the warm up and an empty table Answer the questions below 2 Length of short side cm Length of perimeter cm 0 25 1 2 7 5 6 5 22 3 9 5 0 5 2 1 25 4 5 3 5 12 5 1 5 5 4 14 1 3 5 Length of short side cm Length of perimeter cm 1 How can you organize the measurements from the first table so that any patterns are easier to see Write the organized measurements in the empty table 2 For each of the following lengths estimate the perimeter of an isosceles right triangle whose short sides have that length Explain your reasoning for each triangle a length of short sides is 0 75 cm b length of short sides is 5 cm c 234 length of short sides is 10 cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 1 ACTIVITY 2 3 Here are four scatter plots You will receive four tables of data Match each table with one of the scatter plots Use information from the tables to label the axes for each scatter plot 21 000 45 40 18 000 35 15 000 30 12 000 25 9 000 6 000 20 0 30 000 60 000 90 000 120 000 150 000 15 72 78 84 90 96 102 32 10 000 30 28 8 000 26 24 6 000 22 20 4 000 18 16 2 000 0 9 1 14 1 38 1 62 1 86 2 1 14 1 000 1 250 1 500 1 750 2 000 2 250 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 2 500 235

G8M6 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary Consider the data collected from pulling back a toy car and then letting it go forward In the first table the data may not seem to have an obvious pattern The second table has the same data and shows that both values are increasing together Distance pulled back in Distance traveled in Distance pulled back in Distance traveled in 6 23 57 1 8 95 4 18 48 2 13 86 10 38 66 4 18 48 8 31 12 6 23 57 2 13 86 8 31 12 1 8 95 10 38 66 A scatter plot of the data makes the pattern clear enough that we can estimate how far the car will travel when it is pulled back 5 inches Distance traveled in Patterns in data can sometimes become more obvious when reorganized in a table or when represented in scatter plots or other diagrams If a pattern is observed it can sometimes be used to make predictions 40 30 20 10 0 0 2 4 6 8 10 12 Distance pulled back in TERMINOLOGY Scatter plot 236 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 1 Name Date GRADE 8 MISSION 6 LESSON 1 Exit Ticket Twenty rubber spheres were compressed with varying amounts of force The widths and heights of the resulting shapes were measured Here is a scatter plot that shows the measurements for each sphere 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 Horizontal width inches 1 Label the vertical axis of the scatter plot 2 If a compressed rubber sphere has a 6 inch width is its height closer to 5 inches or to 11 inches Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 237

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ZEARN MATH STUDENT EDITION G8M6 LESSON 2 Lesson 2 Plotting Data Let s collect and display some data about the class Warm Up 1 Lin surveyed 30 students about the longest time they had ever run Andre asked them about their favorite color How could Lin and Andre represent their data sets Would they represent them in the same way Why or why not Concept Exploration ACTIVITY 1 2 Are older students always taller Do taller students tend to have bigger hands To investigate these questions the class will gather data Follow the directions below A person s arm span is the distance between the tips of their index fingers when their arms are fully spread out A person s hand span is the distance from the tip of their thumb to the tip of their little finger when their fingers are fully spread out 1 Each partner should Measure the other partner s height arm span and hand span for their right hand to the nearest centimeter Record the other partner s measurements and age in months in the table Height cm Arm span cm Hand span cm Age months Partner A Partner B 2 One partner records the data from your table in a table of data for the entire class 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 239

G8M6 LESSON 2 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Answer the questions and follow the directions about your data below 1 What types of graphical representations could be used to show the class s height measurements Make a graphical representation of the class s height measurements 2 Choose a color and use it to plot a point on the coordinate plane that represents your own height and hand span Then in the same color plot a second point that represents your partner s height and hand span 3 In a different color plot the height and hand span of each student in your class making a scatter plot of the heights and hand spans for the entire class 4 Based on your scatter plot answer these questions a Do taller students in your class tend to have bigger hands Explain how you know b Is hand span a linear function of height Explain how you know 240 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 2 Lesson Summary Histograms show us how measurements of a single attribute are distributed For example a veterinarian saw 25 dogs in her clinic one week She measured the height and weight of each dog This histogram shows how the weights of the dogs are distributed 8 This histogram shows how the heights of the dogs are distributed 6 7 5 6 4 5 3 4 3 2 2 1 1 0 16 32 48 64 80 96 112 0 6 9 12 15 18 21 24 27 30 Dog height inches Dog weight pounds These histograms tell us how the weights of the dogs and how the heights of dogs were distributed But they do not give any evidence of a connection between a dog s height and its weight Scatter plots allow us to investigate possible connections between two attributes In this example each plotted point corresponds to one of the 25 dogs and its coordinates tell us the height and weight of that dog Examination of the scatter plot allows us to see a connection between height and weight of the dogs Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 241

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ZEARN MATH STUDENT EDITION G8M6 LESSON 2 Name Date GRADE 8 MISSION 6 LESSON 2 Exit Ticket Here is table that shows measurements of right hand length and right foot length for five people 1 Right hand length cm Right foot length cm Person A 19 27 Person B 21 30 Person C 17 23 Person D 18 24 Person E 19 26 Draw a scatter plot for the data 30 right foot length cm 25 20 15 10 5 0 5 10 15 20 25 30 Right hand length cm 2 Circle the point in the scatter plot that represents Person D s measurements 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 243

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ZEARN MATH STUDENT EDITION G8M6 LESSON 3 Lesson 3 What a Point in a Scatter Plot Means Let s investigate points in scatter plots Warm Up A giant panda lives in a zoo What does the point on the graph tell you about the panda Weight in kilograms 1 100 36 82 75 50 25 0 12 24 36 48 60 72 84 Age in months 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 245

G8M6 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Car A 246 The table and scatter plot show weights and fuel efficiencies of 18 cars Weight kg 1 549 Fuel efficiency mpg 25 B 1 610 20 C 1 737 21 D 1 777 20 E 1 486 23 F 1 962 16 G 2 384 16 H 1 957 19 I 2 212 16 J 1 115 29 K 2 068 18 L 1 663 19 M 2 216 18 N 1 432 25 O 1 987 18 P 1 580 26 Q 1 234 30 R 1 656 23 32 Fuel e iciency mpg 2 30 28 26 24 22 20 18 16 14 1 000 1 250 1 500 1 750 2 000 2 250 2 500 Weight kg 1 Which point in the scatter plot represents Car L s measurements 2 What is the fuel efficiency of the car with the greatest weight 3 What is the weight of the car with the greatest fuel efficiency 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 3 4 Car S weighs 1 912 kilograms and gets 16 miles per gallon On the scatter plot plot a point that represents Car S s measurements 5 Cars N and O shown in the scatter plot are made by the same company Compare their weights and fuel efficiencies Does anything surprise you about these cars 6 A different company makes Cars F and G Compare their weights and fuel efficiencies Does anything surprise you about these cars 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 247

G8M6 LESSON 3 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 A clothing store keeps track of the average monthly temperature in degrees Celsius and coat sales in dollars y Temperature degrees Celsius Coat sales dollars 5 1 550 3 1 340 3 1 060 8 1 070 15 680 21 490 23 410 200 21 510 6 4 2 0 100 17 600 200 11 740 6 940 2 1 390 1700 1600 1500 1400 248 Coat sales dollars 1300 1200 1100 1000 900 800 700 600 500 400 300 100 1 2 4 6 8 10 12 14 16 18 20 22 24 26 x Temperature degrees Celsius What does the point 15 680 represent 2 For the month with the lowest average temperature what is the total amount made from coat sales Explain how you used the table to find this information 3 For the month with the smallest coat sales what is the average monthly temperature Explain how you used the scatter plot to find this information 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 3 4 If there were a point at 0 A what would it represent Use the scatter plot to estimate a value for A 5 What would a point at B 0 represent Use the scatter plot to estimate a value for B 6 Would it make sense to use this trend to estimate the value of sales when the average monthly temperature is 60 degrees Celsius Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 249

G8M6 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary Scatter plots show two measurements for each individual from a group For example this scatter plot shows the weight and height for each dog from a group of 25 dogs Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches We can see that the tallest dogs are 27 inches and that one of those tallest dogs weighs about 75 pounds while the other weighs about 110 pounds This shows us that dog weight is not a function of dog height because there would be two different outputs for the same input But we can see a general trend Taller dogs tend to weigh more than shorter dogs There are exceptions For example there is a dog that is 18 inches tall and weighs over 50 pounds and there is another dog that is 21 inches tall but weighs less than 30 pounds When we collect data by measuring attributes like height weight area or volume we call the data numerical data or measurement data and we say that height weight area or volume are numerical variables Upcoming lessons will discuss how to identify and describe trends in collected data 250 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 3 Name Date GRADE 8 MISSION 6 LESSON 3 Exit Ticket Here are a table and scatter plot that show ratings and wins for quarterbacks who started 16 games this season 20 Number of wins A 93 8 4 B 102 2 12 C 93 6 6 D 89 8 E 88 2 5 F 97 7 G 88 7 6 H 91 1 7 I 92 7 10 J 88 10 K 101 6 9 L 104 6 13 M 84 2 6 N 99 4 15 O 110 1 10 P 95 4 11 Q 88 7 11 15 Number of wins Quarterback rating Player 10 5 0 80 90 100 110 120 Quarterback rating 1 Circle the point in the scatter plot that represents Player K s data 2 Which quarterback s data are represented by the point farthest to the left 3 Player R is not included in the table because he did not start 16 games this year He did have a quarterback rating of 99 4 and his team won 8 games On the scatter plot plot a point that represents Player R s data 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 251

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ZEARN MATH STUDENT EDITION G8M6 LESSON 4 Lesson 4 Fitting a Line to Data Let s look at the scatter plots as a whole Warm Up 1 Here is a scatter plot that shows the weights and fuel efficiencies of 20 different types of cars If a car weighs 1 750 kg would you expect its fuel efficiency to be closer to 22 mpg or to 28 mpg Explain your reasoning Fuel e iciency mpg 32 30 28 26 24 22 20 18 16 14 1 000 1 250 1 500 1 750 2 000 2 250 2 500 Weight kg 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 253

G8M6 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 254 The table below and the graph on the next page show the weights and prices of 20 different diamonds Graphed on the scatter plot is the line y 5 520x 1 091 Use this information to answer the questions Weight carats Actual price dollars Predicted price dollars 1 3 772 4 429 1 4 221 4 429 1 4 032 4 429 1 5 385 4 429 1 05 3 942 4 705 1 05 4 480 4 705 1 06 4 511 4 760 1 2 5 544 5 533 1 3 6 131 6 085 1 32 5 872 6 195 1 41 7 122 6 692 1 5 7 474 7 189 1 5 5 904 7 189 1 59 8 706 7 686 1 61 8 252 7 796 1 73 9 530 8 459 1 77 9 374 8 679 1 85 8 169 9 121 1 9 9 541 9 397 2 04 9 125 10 170 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 4 The function described by the equation y 5 520x 1 091 is a model of the relationship between a diamond s weight and its price This model predicts the price of a diamond from its weight These predicted prices are shown in the third column of the table 11 000 Price dollars 10 000 9 000 8 000 7 000 6 000 5 000 4 000 3 000 2 000 0 9 1 02 1 14 1 26 1 38 1 5 1 62 1 74 1 86 1 98 2 1 Weight carats 1 Two diamonds that both weigh 1 5 carats have different prices What are their prices How can you see this in the table How can you see this in the graph 2 The model predicts that when the weight is 1 5 carats the price will be 7 189 How can you see this in the graph How can you see this using the equation 3 One of the diamonds weighs 1 9 carats What does the model predict for its price How does that compare to the actual price 4 Find a diamond for which the model makes a very good prediction of the actual price How can you see this in the table In the graph 5 Find a diamond for which the model s prediction is not very close to the actual price How can you see this in the table In the graph 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 255

G8M6 LESSON 4 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Use this scatter plot that shows the lengths and widths of 20 different left feet to answer the questions 12 Foot width cm 11 10 9 8 7 20 22 24 26 28 30 32 Foot length cm 1 Estimate the widths of the longest foot and the shortest foot 2 Estimate the lengths of the widest foot and the narrowest foot 3 Here is the same scatter plot together with the graph of a model for the relationship between foot length and width 12 Foot width cm 11 10 9 8 7 20 22 24 26 28 30 32 Foot length cm Circle the data point that is far away from the other points and doesn t fit the trend What length and width does that point represent 256 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 4 Lesson Summary Sometimes we can use a linear function as a model of the relationship between two variables For example here is a scatter plot that shows heights and weights of 25 dogs together with the graph of a linear function which is a model for the relationship between a dog s height and its weight Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches We can see that the model does a good job of predicting the weight given the height for some dogs These correspond to points on or near the line The model doesn t do a very good job of predicting the weight given the height for the dogs whose points are far from the line For example there is a dog that is about 20 inches tall and weighs a little more than 16 pounds The model predicts that the weight would be about 48 pounds We say that the model overpredicts the weight of this dog There is also a dog that is 27 inches tall and weighs about 110 pounds The model predicts that its weight will be a little less than 80 pounds We say the model underpredicts the weight of this dog Sometimes a data point is far away from the other points or doesn t fit a trend that all the other points fit We call these outliers TERMINOLOGY outlier 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 257

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ZEARN MATH STUDENT EDITION G8M6 LESSON 4 Name Date GRADE 8 MISSION 6 LESSON 4 Exit Ticket Below is a scatter plot that shows the lengths and widths of 20 left feet together with the graph of a model of the relationship between foot length and width Use this information to answer the questions 12 Foot width cm 11 10 9 8 7 20 22 24 26 28 30 32 Foot length cm 1 Draw a box around the point that represents the foot with length closest to 29 cm 2 What is the approximate width of this foot 3 What width does the model predict for a foot with length 29 cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 259

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ZEARN MATH STUDENT EDITION G8M6 LESSON 5 Lesson 5 Describing Trends in Scatter Plots Let s look for associations between variables Warm Up 1 Which one doesn t belong 80 50 60 25 40 0 20 25 0 0 3 6 9 12 50 0 100 80 75 60 50 40 25 20 0 0 3 6 9 12 0 0 3 6 9 12 3 6 9 12 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 261

G8M6 LESSON 5 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 40 40 30 30 20 20 10 10 0 0 3 3 6 9 0 12 40 30 30 20 20 10 10 0 3 43 0 3 6 9 12 Here are two copies of another scatter plot Use the pasta and straightedge to experiment with drawing lines to fit the data Pick the line that you think best fits the data Compare it with a partner s 40 0 262 Here are two copies of the same scatter plot You will receive a piece of pasta and a straightedge Experiment with drawing lines to fit the data Pick the line that you think best fits the data Compare it with a partner s 6 9 12 0 0 3 6 9 12 In your own words describe what makes a line fit a data set well 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 5 ACTIVITY 2 Is this line a good fit for the data Explain your reasoning 53 4 000 3 000 2 000 1 000 0 1 000 1 125 1 250 1 375 1 500 Draw a line that fits the data better 63 4 000 3 000 2 000 1 000 0 1 000 73 1 125 1 250 1 375 1 500 Is this line a good fit for the data Explain your reasoning 200 150 100 50 0 0 25 50 75 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 263

G8M6 LESSON 5 ZEARN MATH STUDENT EDITION Draw a line that best fits the data 83 200 150 100 50 0 264 0 25 50 75 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 5 Lesson Summary When a linear function fits data well we say there is a linear association between the variables For example the relationship between height and weight for 25 dogs with the linear function whose graph is shown in the scatter plot Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches Because the model fits the data well and because the slope of the line is positive we say that there is a positive association between dog height and dog weight What do you think the association between the weight of a car and its fuel efficiency is Fuel e iciency mpg 32 30 28 26 24 22 20 18 16 14 1 000 1 250 1 500 1 750 2 000 2 250 2 500 Weight kg Because the slope of a line that fits the data well is negative we say that there is a negative association between the fuel efficiency and weight of a car TERMINOLOGY negative association positive association 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 265

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ZEARN MATH STUDENT EDITION G8M6 LESSON 5 Name Date GRADE 8 MISSION 6 LESSON 5 Exit Ticket 1 Elena said I think this line is a good fit because half of the points are on one side of the line and half of the points are on the other side Do you agree Explain your reasoning 80 60 40 20 0 2 0 2 4 6 8 10 12 Noah said I think this line is a good fit because it passes through the leftmost point and the rightmost point Do you agree Explain your reasoning 80 60 40 20 0 0 2 4 6 8 10 12 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 267

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ZEARN MATH STUDENT EDITION G8M6 LESSON 6 Lesson 6 The Slope of a Fitted Line Let s look at how changing one variable changes another Warm Up 1 Estimate the slope of the line y 10 5 5 2 10 5 0 5 8 3 10 x 5 4 6 10 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 269

G8M6 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 For each scatter plot decide if there is an association between the two variables and describe the situation using one of these sentences For these data as increases tends to increase For these data as increases tends to decrease For these data and do not appear to be related 21 000 1 18 000 Price 15 000 12 000 9 000 6 000 0 30 000 60 000 90 000 Mileage 120 000 150 000 11 000 2 10 000 Price dollars 9 000 8 000 7 000 6 000 5 000 4 000 3 000 2 000 0 9 1 02 1 14 1 26 1 38 1 5 1 62 1 74 1 86 1 98 2 1 Weight carats 45 Energy consumed kwh 3 40 35 30 25 20 15 70 75 80 85 90 95 100 105 High temperature F 270 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 6 ACTIVITY 2 3 For each of the situations a linear model for some data is shown Answer the questions for each situation 1 What is the slope of the line in the scatter plot for each situation 2 What is the meaning of the slope in that situation y 5 520 619x 1 091 393 11 000 10 000 Price dollars 9 000 8 000 7 000 6 000 5 000 4 000 3 000 2 000 0 9 1 02 1 14 1 26 1 38 1 5 1 62 1 74 1 86 1 98 2 1 Weight carats Fuel e iciency mpg y 0 011x 40 604 32 30 28 26 24 22 20 18 16 14 1 000 1 250 1 500 1 750 2 000 2 250 2 500 Weight kg y 0 59x 21 912 Energy consumed kwh 45 40 35 30 25 20 15 70 75 80 85 90 95 100 105 High temperature F 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 271

G8M6 LESSON 6 ZEARN MATH STUDENT EDITION ACTIVITY 3 43 Use the scatter plots to answer the questions 1 For each of the scatter plots decide whether it makes sense to fit a linear model to the data If it does would the graph of the model have a positive slope a negative slope or a slope of zero A 60 20 40 10 20 0 3 6 9 0 12 D 50 0 3 6 9 12 00 15 30 45 60 80 25 60 0 40 25 20 50 0 E 80 30 0 C B 40 3 6 9 12 12 11 10 9 8 720 22 24 26 28 30 32 2 Which of the scatter plots show evidence of a positive association between the variables Of a negative association Which do not appear to show an association 272 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 6 Lesson Summary Here is a scatter plot that we have seen before As noted earlier we can see from the scatter plot that taller dogs tend to weigh more than shorter dogs Another way to say it is that weight tends to increase as height increases When we have a positive association between two variables an increase in one means there tends to be an increase in the other Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches We can quantify this tendency by fitting a line to the data and finding its slope For example the equation of the fitted line is w 4 27h 37 where h is the height of the dog and w is the predicted weight of the dog Dog weight pounds 112 96 80 64 48 32 16 0 6 9 12 15 18 21 24 27 30 Dog height inches The slope is 4 27 which tells us that for every 1 inch increase in dog height the weight is predicted to increase by 4 27 pounds 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 273

G8M6 LESSON 6 ZEARN MATH STUDENT EDITION Fuel e iciency mpg In our example of the fuel efficiency and weight of a car the slope of the fitted line shown is 0 01 32 30 28 26 24 22 20 18 16 14 1 000 1 250 1 500 1 750 2 000 2 250 2 500 Weight kg This tells us that for every 1 kilogram increase in the weight of the car the fuel efficiency is predicted to decrease by 0 01 miles per gallon When we have a negative association between two variables an increase in one means there tends to be a decrease in the other 274 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 6 Name Date GRADE 8 MISSION 6 LESSON 6 Exit Ticket Here is a scatter plot that shows the years when some used cars were made and their prices in 2016 together with the graph of a linear model for the relationship between year and price 21 000 Price 18 000 15 000 12 000 9 000 6 000 2006 2008 2010 2012 Year 2014 2016 1 Is the slope positive or negative 2 Which of these values is closest to the slope of the linear model shown in the scatter plot a 1 000 b 3 000 c 1 000 d 3 000 3 Use the value you selected to describe the meaning of the slope in this context 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 275

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ZEARN MATH STUDENT EDITION G8M6 LESSON 7 Lesson 7 Observing More Patterns in Scatter Plots Let s look for other patterns in data Warm Up What do you notice What do you wonder Crude oil import price dollars per barrel 1 100 80 60 40 20 0 250 000 300 000 350 000 400 000 450 000 Oil production thousands of tons Concept Exploration ACTIVITY 1 2 You will each get a set of cards Each card shows a scatter plot 1 Sort the cards into categories and describe each category 2 Explain the reasoning behind your categories to your partner Listen to your partner s reasoning for their categories 3 Sort the cards into two categories positive associations and negative associations Compare your sorting with your partner s and discuss any disagreements 4 Sort the cards into two categories linear associations and non linear associations Compare your sorting with your partner s and discuss any disagreements 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 277

G8M6 LESSON 7 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 How are these scatter plots alike How are they different A 40 38 40 25 80 13 6 12 18 00 24 D 30 40 0 20 15 0 3 5 8 10 3 5 8 10 6 12 18 24 60 15 300 278 50 0 1200 C B 200 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 7 Lesson Summary Sometimes a scatter plot shows an association that is not linear 30 24 18 12 6 0 0 6 3 9 12 We call such an association a non linear association In later grades you will study functions that can be models for non linear associations Sometimes in a scatter plot we can see separate groups of points 50 50 38 38 25 25 13 13 00 3 5 8 10 00 3 5 8 10 We call these groups clusters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 279

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ZEARN MATH STUDENT EDITION Name G8M6 LESSON 7 Date GRADE 8 MISSION 6 LESSON 7 Exit Ticket 1 Draw a scatter plot that shows a positive linear association and clustering 2 Draw a scatter plot that shows a negative non linear association and no clustering 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 281

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ZEARN MATH STUDENT EDITION G8M6 LESSON 8 Lesson 8 Analyzing Bivariate Data Let s analyze data like a pro Warm Up 1 A researcher found an association between a dog s stride length and its speed The longer a dog s steps the faster it goes The predicted speed in meters per second s as a function of step length in meters l is s 4l 1 6 What does the rate of change of the function tell you about the association between stride length and speed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 283

G8M6 LESSON 8 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Is there an association between the weight of an animal s body and the weight of the animal s brain Use the data in the table to make a scatter plot Are there any outliers Then answer the questions on the next page Body weight kg Brain weight g 1 800 cow 465 423 1 200 grey wolf 36 120 goat 28 115 donkey 187 419 horse 521 284 1 600 1 400 Brain weight g Animal 1 000 800 600 400 200 655 potar monkey 10 115 cat 3 26 giraffe 529 680 gorilla 207 406 human 62 1320 resus monkey 7 179 kangaroo 35 56 sheep 56 175 jaguar 100 157 chimpanzee 52 440 pig 192 180 0 0 100 200 300 400 500 600 Body weight kg 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 8 1 After removing the outliers does there appear to be an association between body weight and brain weight Describe the association in a sentence 2 Using a piece of pasta as a straightedge fit a line to your scatter plot and estimate its slope What does this slope mean in the context of brain and body weight 3 Does the fitted line help you identify more outliers 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 285

G8M6 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary People often collect data in two variables to investigate possible associations between two numerical variables and use the connections that they find to predict more values of the variables Data analysis usually follows these steps 1 Collect data 2 Organize and represent the data and look for an association 3 Identify any outliers and try to explain why these data points are exceptions to the trend that describes the association 4 Find a function that fits the data well Although computational systems can help with data analysis by graphing the data finding a function that might fit the data and using that function to make predictions it is important to understand the process and think about what is happening A computational system may find a function that does not make sense or use a line when the situation suggests that a different model would be more appropriate 286 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G8M6 LESSON 8 Date GRADE 8 MISSION 6 LESSON 8 Exit Ticket 1 Draw a line on the scatter plot that fits the data well 2 A new point will be added to the scatter plot with x 4 What do you predict for the y value of this point 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 287

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ZEARN MATH STUDENT EDITION G8M6 LESSON 9 Lesson 9 Looking for Associations Let s look for associations in data Warm Up 1 What do you notice What do you wonder 40 Watches TV 30 Not much TV 20 10 0 Plays sports No sports Concept Exploration ACTIVITY 1 2 Use the cards you receive and follow the directions below Some cards show two way tables like this Has cell phone Does not have cell phone Total 10 to 12 years old 25 35 60 13 to 15 years old 40 10 50 16 to 18 years old 50 10 60 Total 115 55 170 Some cards show bar graphs like this 60 Has cell phone 45 No cell phone 30 15 0 10 12 years old 13 15 years old 16 18 years old 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 289

G8M6 LESSON 9 ZEARN MATH STUDENT EDITION Some cards show segmented bar graphs like this 100 Has cell phone 75 No cell phone 50 25 0 10 12 years old 13 15 years old 16 18 years old The bar graphs and segmented bar graphs have their labels removed 290 1 Put all the cards that describe the same situation in the same group 2 One of the groups does not have a two way table Make a two way table for the situation described by the graphs in the group 3 Label the bar graphs and segmented bar graphs so that the categories represented by each bar are indicated 4 Describe in your own words the kind of information shown by a segmented bar graph 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 9 ACTIVITY 2 3 1 Here is a two way table that shows data about cell phone usage among children aged 10 to 18 Has cell phone Does not have cell phone Total 10 to 12 years old 25 35 60 13 to 15 years old 40 10 50 16 to 18 years old 50 10 60 Total 115 55 170 Complete the table In each row the entries for has cell phone and does not have cell phone should have the total 100 Round entries to the nearest percentage point Has cell phone 10 to 12 years old Does not have cell phone 42 13 to 15 years old 16 to 18 years old Total 100 17 This is still a two way table Instead of showing frequency this table shows relative frequency 2 Two way tables that show relative frequencies often don t include a total row at the bottom Why 3 Is there an association between age and cell phone use How does the two way table of relative frequencies help to illustrate this 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 291

G8M6 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary When we collect data by counting things in various categories like red blue or yellow we call the data categorical data and we say that color is a categorical variable We can use two way tables to investigate possible connections between two categorical variables For example this two way table of frequencies shows the results of a study of meditation and state of mind of athletes before a track meet Meditated Did not meditate Total Calm 45 8 53 Agitated 23 21 44 Total 68 29 97 If we are interested in the question of whether there is an association between meditating and being calm we might present the frequencies in a bar graph grouping data about meditators and grouping data about non meditators so we can compare the numbers of calm and agitated athletes in each group Calm 50 Agitated 40 30 20 10 0 Meditated Did not meditate Notice that the number of athletes who did not meditate is small compared to the number who meditated 29 as compared to 68 as shown in the table If we want to know the proportions of calm meditators and calm non meditators we can make a two way table of relative frequencies and present the relative frequencies in a segmented bar graph 292 Meditated Did not meditate Calm 66 28 Agitated 34 72 Total 100 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 9 Calm 100 Agitated 75 50 25 0 Meditated Did not meditate TERMINOLOGY Relative frequency Segmented bar graph Two way table 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 293

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ZEARN MATH STUDENT EDITION G8M6 LESSON 9 Name Date GRADE 8 MISSION 6 LESSON 9 Exit Ticket 1 In a class of 25 students some students play a sport some play a musical instrument some do both some do neither Complete the two way table to show data that might come from this class Plays an instrument Does not play an instrument Total Plays a sport Does not play a sport 5 Total 2 25 Using the entries from the previous table complete this table so that it shows relative frequencies Round entries to the nearest percentage point Plays an instrument Does not play an instrument Total Plays a sport Does not play a sport 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 295

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ZEARN MATH STUDENT EDITION G8M6 LESSON 10 Lesson 10 Using Data Displays to Find Associations Let s use data displays to find associations Warm Up 1 1 For a survey students in a class answered these questions Do you play a sport Do you play a musical instrument Here is a two way table that gives some results from the survey Complete the table assuming that all students answered both questions Plays instrument Plays sport Does not play instrument 5 Total 16 Does not play sport Total 15 25 2 To the nearest percentage point what percentage of students who play a sport don t play a musical instrument 3 To the nearest percentage point what percentage of students who don t play a sport also don t play a musical instrument 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 297

G8M6 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 You will receive a two way table with information about the number of people in our class who play sports or musical instruments Complete this table to make a two way table for the data you received The table will show relative frequencies by row Plays instrument 2 Does not play instrument Row total Plays sport 100 Does not play sport 100 Make a segmented bar graph for the table Use one bar of the graph for each row of the table 100 75 50 25 0 298 Plays sports No sports 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 G8M6 LESSON 10 Complete the table to make a two way table for the data you received The table will show relative frequencies by column Plays instrument Does not play instrument 100 100 Plays sport Does not play sport Column total 4 Using the values in the table make a segmented bar graph Use one bar of the graph for each column of the table 100 75 50 25 0 5 Plays instrument No instrument Based on the two way tables and segmented bar graphs do you think there is an association between playing a sport and playing a musical instrument Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 299

G8M6 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 An eraser factory has five machines One machine makes the eraser shapes Then each shape goes through the red machine blue machine yellow machine or green machine to have a side colored The manager notices that an uncolored side of some erasers is flawed at the end of the process and wants to know which machine needs to be fixed the shape machine or some of the color machines The manager collected data on the number of flawed and unflawed erasers of each color 300 Unflawed Flawed Total Red 285 15 300 Blue 223 17 240 Yellow 120 80 200 Green 195 65 260 Total 823 177 1000 1 Work with a partner Each of you should make one segmented bar graph for the data in the table One segmented bar graph should have a bar for each row of the table The other segmented bar graph should have one bar for each column of the table 2 Are the flawed erasers associated with certain colors If so which colors Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 10 Lesson Summary In an earlier lesson we looked at data on meditation and state of mind in athletes Calm 50 Agitated 40 30 20 10 0 Meditated Did not meditate Calm 100 Agitated 75 50 25 0 Meditated Did not meditate Is there an association between meditation and state of mind The bar graph shows that more athletes were calm than agitated among the group that meditated and more athletes were agitated than calm among the group that did not We can see the proportions of calm meditators and calm non meditators from the segmented bar graph which shows that about 66 of athletes who meditated were calm whereas only about 27 of those who did not meditate were calm This does not necessarily mean that meditation causes calm it could be the other way around that calm athletes are more inclined to meditate But it does suggest that there is an association between meditating and calmness 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 301

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ZEARN MATH STUDENT EDITION G8M6 LESSON 10 Name Date GRADE 8 MISSION 6 LESSON 10 Exit Ticket Here are a two way table and segmented bar graph for data from students in two classes Do they show evidence of differences between the two classes Prefers math Prefers science Prefers recess Class A 6 3 8 Class B 8 7 15 100 Recess 75 Science Math 50 25 0 Class A Class B 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 303

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ZEARN MATH STUDENT EDITION G8M6 LESSON 11 Lesson 11 Gone in 30 seconds Let s gather and analyze some timing data Concept Exploration ACTIVITY 1 1 1 2 In this activity you ll get two chances to guess how long 30 seconds is Then look for an association between the two guesses of all the students in your class Work with a partner Follow the instructions listed here to gather your data One of you will hold a stopwatch where the other person cannot see it The person holding the stopwatch says go and starts the timer The other person says stop when they think 30 seconds have passed The person holding the stopwatch will stop the timer then report and record the time to the nearest second The person holding the stopwatch will give a second chance repeating the experiment After both times are recorded switch roles Record the group data in this table When you finish a group member should give the data to the teacher Name 3 Time 1 Time 2 Look at your data Comparing Time 1 to Time 2 do you think there is a positive association a negative association or no association Discuss your thinking with your group 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 305

G8M6 LESSON 11 ZEARN MATH STUDENT EDITION 4 What are some ways you could organize and represent the entire class s data 5 Make a scatter plot of the entire class s data and look for patterns Identify any outliers and the type of any association you observe 6 Draw two lines on your scatter plot a vertical line and a horizontal line each representing 30 seconds for one trial Use the table for the class s data to complete this two way table Time 2 30 sec Time 2 30 sec Time 2 30 sec Total Time 1 30 sec Time 1 30 sec Time 1 30 sec Total 7 306 Use the two way table to decide whether there is an association between Time 1 and Time 2 Explain how you know 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 12 Lesson 12 Keeping Track of All Possible Outcomes Let s explore sample spaces for experiments with multiple parts Warm Up 1 How many different meals are possible if each meal includes one main course one side dish and one drink Main courses Side dishes Drinks grilled chicken salad milk turkey sandwich applesauce juice pasta salad water 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 307

G8M6 LESSON 12 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Consider the experiment Flip a coin and then roll a number cube Elena Kiran and Priya each use a different method for finding the sample space of this experiment Read about each method below and then answer the questions that follow Elena carefully writes a list of all the options Heads 1 Heads 2 Heads 3 Heads 4 Heads 5 Heads 6 Tails 1 Tails 2 Tails 3 Tails 4 Tails 5 Tails 6 Kiran makes a table 1 2 3 4 5 6 H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6 Priya draws a tree with branches in which each pathway represents a different outcome H 1 2 3 4 5 6 T 1 2 3 4 5 6 1 Compare the three methods What is the same about each method What is different Be prepared to explain why each method produces all the different outcomes without repeating any 2 Which method do you prefer for this situation Pause here so your teacher can review your work 308 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 G8M6 LESSON 12 Find the sample space for each of these experiments using any method Make sure you list every possible outcome without repeating any a Flip a dime then flip a nickel and then flip a penny Record whether each lands heads or tails up b Han s closet has a blue shirt a gray shirt a white shirt blue pants khaki pants and black pants He must select one shirt and one pair of pants to wear for the day c Spin a color and then spin a number d Spin the hour hand on an analog clock and then choose a m or p m 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 309

G8M6 LESSON 12 ZEARN MATH STUDENT EDITION ACTIVITY 2 43 A submarine sandwich shop makes sandwiches with one kind of bread one protein one choice of cheese and two vegetables as shown below Answer the questions Breads Italian white wheat Proteins Tuna ham turkey beans Cheese Provolone Swiss American none Vegetables Lettuce tomatoes peppers onions pickles Submarine sandwich and chips by jeffreyw via Wikimedia Commons Public Domain 310 1 How many different sandwiches are possible Explain your reasoning You do not need to write out the sample space 2 Andre knows he wants a sandwich that has ham lettuce and tomatoes on it He doesn t care about the type of bread or cheese How many of the different sandwiches would make Andre happy 3 If a sandwich is made by randomly choosing each of the options what is the probability it will be a sandwich that Andre would be happy with 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 12 Lesson Summary Sometimes we need a systematic way to count the number of outcomes that are possible in a given situation For example suppose there are 3 people A B and C who want to run for the president of a club and 4 different people 1 2 3 and 4 who want to run for vice president of the club We can use a tree a table or an ordered list to count how many different combinations are possible for a president to be paired with a vice president With a tree we can start with a branch for each of the people who want to be president Then for each possible president we add a branch for each possible vice president for a total of 3 4 12 possible pairs We can also start by counting vice presidents first and then adding a branch for each possible president for a total of 4 3 12 possible pairs AA 11 11 22 33 44 AA CC AA 22 11 33 44 AA 33 BB CC 11 22 33 44 CC BB CC 22 BB BB AA 44 BB CC A table can show the same result 1 2 3 4 A A 1 A 2 A 3 A 4 B B 1 B 2 B 3 B 4 C C 1 C 2 C 3 C 4 So does this ordered list A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 311

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ZEARN MATH STUDENT EDITION Name G8M6 LESSON 12 Date GRADE 8 MISSION 6 LESSON 12 Exit Ticket Andre is reviewing proportional relationships He wants to practice using a graph that goes through a point so that each coordinate is between 1 and 10 1 For the point how many outcomes are in the sample space 2 For how many outcomes are the x coordinate and the y coordinate the same number 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 313

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ZEARN MATH STUDENT EDITION G8M6 LESSON 13 Lesson 13 Multi step Experiments Let s look at probabilities of experiments that have multiple steps Warm Up Is each equation true or false Explain your reasoning 8 8 8 8 8 3 10 10 10 10 10 5 10 6 4 6 3 6 4 6 5 Concept Exploration ACTIVITY 1 1 The other day you wrote the sample space for spinning each of these spinners once What is the probability of getting 1 Green and 3 2 Blue and any odd number 3 Any color other than red and any number other than 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 315

G8M6 LESSON 13 ZEARN MATH STUDENT EDITION ACTIVITY 2 1 2 Write the sample space for flipping a coin and rolling a number cube using the representation you are assigned 3 You looked at a list a table and a tree that showed the sample space for rolling a number cube and flipping a coin Answer the questions in your notes Your teacher will assign you one of these three structures to use to answer these questions Be prepared to explain your reasoning a What is the probability of getting tails and a 6 b What is the probability of getting heads and an odd number Pause here so your teacher can review your work 2 Suppose you roll two number cubes What is the probability of getting a Both cubes showing the same number b Exactly one cube showing an even number c At least one cube showing an even number d Two values that have a sum of 8 e Two values that have a sum of 13 3 316 Jada flips three quarters What is the probability that all three will land showing the same side 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G8M6 LESSON 13 Lesson Summary Suppose we have two bags One contains 1 star block and 4 moon blocks The other contains 3 star blocks and 1 moon block If we select one block at random from each what is the probability that we will get two star blocks or two moon blocks To answer this question we can draw a tree diagram to see all of the possible outcomes star star star moon star moon star moon star star star moon star star moon moon star moon star star star star star moon moon There are 5 4 20 possible outcomes Of these 3 of them are both stars and 4 are both moons So the 7 probability of getting 2 star blocks or 2 moon blocks is 20 In general if all outcomes in an experiment are equally likely then the probability of an event is the fraction of outcomes in the sample space for which the event occurs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 317

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ZEARN MATH STUDENT EDITION Name G8M6 LESSON 13 Date GRADE 8 MISSION 6 LESSON 13 Exit Ticket Lin plays a game that involves a standard number cube and a deck of ten cards numbered 1 through 10 If both the cube and card have the same number Lin gets another turn Otherwise the game continues with the next player What is the probability that Lin gets another turn 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 319

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ZEARN MATH STUDENT EDITION G8V2 Terminology Constant term In an expression like 5x 2 the number 2 is called the constant term because it doesn t change when x changes In the expression 7x 9 9 is the constant term In the expression 5x 8 8 is the constant term In the expression 12 4x 12 is the constant term Dependent variable A dependent variable represents the output of a function We need to buy 20 pieces of fruit and decide to buy apples and bananas If we select the number of apples first the equation b 20 a shows the number of bananas we can buy The number of bananas is the dependent variable because it depends on the number of apples Function A function is a rule that assigns exactly one output to each possible input The function y 6x 4 assigns one value of the output y to each value of the input x For example when x 5 then y 6 5 4 or 34 Independent variable An independent variable is an amount that is used to calculate another amount An independent variable represents the input of a function We need to buy 20 pieces of fruit and decide to buy some apples and bananas If we select the number of apples first the equation b 20 a shows the number of bananas we can buy The number of apples is the independent variable because we can choose any number for it Negative association Outlier 120 80 40 6 8 10 12 14 28 30 32 Price in dollars 12 11 Foot width cm An outlier is a data value that is far from the other values in the data set Here is a scatter plot that shows lengths and widths of 20 different left feet The foot whose length is 24 2 cm and width is 7 4 cm is an outlier 160 Number sold A negative association is a relationship between two quantities where one tends to decrease as the other increases In a scatter plot the data points tend to cluster around a line with negative slope Different stores across the country sell a book for different prices The scatter plot shows that there is a negative association between the price of the book in dollars and the number of books sold at that price 10 9 8 7 20 22 24 26 Foot length cm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 321

G8V2 ZEARN MATH STUDENT EDITION Positive association Dog weight pounds A positive association is a relationship between two quantities where one tends to increase as the 112 other increases In a scatter plot the data points tend to 96 cluster around a line with positive slope 80 The relationship between height and weight for 25 64 dogs is shown in the scatter plot There is a positive 48 32 association between dog height and dog weight 16 0 Relative frequency 6 9 12 15 18 21 Dog height inches 24 27 30 The relative frequency of a category tells us the proportion at which the category occurs in the data set It is displayed as a fraction or a percentage of the total number There were 21 dogs in the park some white some brown some black and some multi color The table shows the frequency and the relative frequency of each color The relative frequency can also be expressed as a decimal or a percentage Color Frequency Relative frequency White 5 5 21 24 Brown 7 7 21 33 Black 3 3 21 14 Multi color 6 6 21 29 112 A scatter plot is a graph that shows the values of two variables on a coordinate plane It allows us to investigate connections between the two variables Each plotted point corresponds to one of 25 dogs The coordinates of each point tell us the height and weight of that dog 322 Dog weight pounds Scatter plot 96 80 64 48 32 16 0 6 9 12 15 18 21 Dog height inches 24 27 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 30

ZEARN MATH STUDENT EDITION G8V2 Segmented bar graph A segmented bar graph compares two categories 100 Has cell phone within a data set The whole bar represents all 75 No cell phone the data within one category Then each bar is 50 separated into parts segments that show the 25 percentage of each part in the second category 0 This segmented bar graph shows the percentage 10 12 13 15 16 18 years old years old years old of people in different age groups that do and do not have a cell phone For example among people ages 10 to 12 about 40 have a cell phone and 60 do not have a cell phone System of equations A system of equations is a set of two or more equations Each equation contains two or more variables We want to find values for the variables that make all the equations true These equations make up a system of equations x y 2 x y 12 The solution to this system is x 5 and y 7 because when these values are substituted for x and y each equation is true 5 7 2 and 5 7 12 Two way table A two way table provides a way to compare two categorical variables It shows one of the variables across the top and the other down one side Each entry in the table is the frequency or relative frequency of the category shown by the column and row headings A study investigates the connection between meditation and the state of mind of athletes before a track meet This two way table shows the results of the study Meditated Did not meditate Total Calm 45 8 53 Agitated 23 21 44 Total 68 29 97 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 323

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zearn org NAME Grade 8 Student Edition Vol 1 Mission 1 Rigid Transformations and Congruence Mission 2 Dilations Similarity and Introducing Slope Mission 3 Linear Relationships Mission 4 Linear Equations and Linear Systems Mission 5 Functions and Volume Mission 6 Associations in Data Student Edition Vol 2 Vol 3 Mission 7 Exponents and Scientific Notation Mission 8 Pythagorean Theorem and Irrational Numbers Mission 9 Putting It All Together G8 Vol 2 Zearnmath_SE_Grade8_Vol2 indd 1 Grade 8 Volume 2 MISSIONS 1 2 3 4 5 6 7 8 9 12 16 22 2 58 PM