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STUDENT EDITION Grade 8 VOLUME 1 Mission 1 Rigid Transformations and Congruence Mission 2 Dilations Similarity and Introducing Slope Mission 3 Linear Relationships 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 Portions of this document are 2019 Illustrative Mathematics and subject to a Creative Commons Attribution Commercial 4 0 License CC BY 4 0 https creativecommons org licenses by 4 0 Zearn is a registered trademark Printed in the U S A ISBN 979 8 88868 883 0

Table of Contents Mission 1 Lesson 1 Moving in the Plane 3 Lesson 2 Naming the Moves 7 Lesson 3 Grid Moves 13 Lesson 4 Making the Moves 19 Lesson 5 Coordinate Moves 25 Lesson 6 Describing Transformations 33 Lesson 7 No Bending or Stretching 39 Lesson 8 Rotation Patterns 45 Lesson 9 Moves in Parallel 51 Lesson 10 Composing Figures 59 Lesson 11 What Is the Same 65 Lesson 12 Congruent Polygons 71 Lesson 13 Congruence 81 Lesson 14 Alternate Interior Angles 87 Lesson 15 Adding the Angles in a Triangle 95 Lesson 16 Parallel Lines and the Angles in a Triangle 101 Lesson 17 Rotate and Tessellate 109 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 2 Lesson 1 Projecting and Scaling 115 Lesson 2 Circular Grid 121 Lesson 3 Dilations with no Grid 127 Lesson 4 Dilations on a Square Grid 135 Lesson 5 More Dilations 141 Lesson 6 Similarity 145 Lesson 7 Similar Polygons 151 Lesson 8 Similar Triangles 157 Lesson 9 Side Length Quotients in Similar Triangles 163 Lesson 10 Meet Slope 169 Lesson 11 Writing Equations for Lines 175 Lesson 12 Using Equations for Lines 181 Lesson 13 The Shadow Knows 187 Mission 3 iv Lesson 1 Understanding Proportional Relationships 193 Lesson 2 Graphs of Proportional Relationships 201 Lesson 3 Representing Proportional Relationships 207 Lesson 4 Comparing Proportional Relationships 213 Lesson 5 Introduction to Linear Relationships 221 Lesson 6 More Linear Relationships 227 Lesson 7 Representations of Linear Relationships 233 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

Lesson 8 Translating to y mx b 241 Lesson 9 Slopes Don t Have to be Positive 247 Lesson 10 Calculating Slope 255 Lesson 11 Equations of All Kinds of Lines 261 Lesson 12 Solutions to Linear Equations 267 Lesson 13 More Solutions to Linear Equations 273 Lesson 14 Using Linear Relations to Solve Problems 279 Lesson 15 Linear Inequalities in Two Variables 283 Terminology 291 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 1 Rigid Transformations and Congruence

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ZEARN MATH STUDENT EDITION G8M1 LESSON 1 Lesson 1 Moving in the Plane Let s describe ways figures can move in the plane Warm Up Which one doesn t belong 1 A B C D Concept Exploration ACTIVITY 1 2 1 Your teacher will give you three pictures Each shows a different set of dance moves Arrange the three pictures so you and your partner can both see them the right way up Choose who will start the game The starting player mentally chooses A B or C and describes the dance to the other player The other player identifies which dance is being talked about A B or C 2 After one round trade roles When you have described all three dances come to an agreement on the words you use to describe the moves in each dance 3 With your partner write a description of the moves in each dance 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

G8M1 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary Here are two ways of changing the position of a figure in a plane without changing its shape or size 4 Sliding or shifting the figure without turning it Shifting Figure A to the right and up puts it in the position of Figure B Turning or rotating the figure around a point Figure A is rotated around the bottom vertex to create Figure C B A A C 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 G8M1 LESSON 1 Name Date GRADE 8 MISSION 1 LESSON 1 Exit Ticket Here are successive positions of a shape Frame 1 Frame 2 Frame 3 Frame 4 Describe how the shape moves from 1 Frame 1 to Frame 2 2 Frame 2 to Frame 3 3 Frame 3 to Frame 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 5

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ZEARN MATH STUDENT EDITION G8M1 LESSON 2 Lesson 2 Naming the Moves Let s be more precise about describing moves of figures in the plane Warm Up 1 Quadrilateral A can be rotated into the position of Quadrilateral B Estimate the angle of rotation 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

G8M1 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Here is another set of dance moves Describe each move or say if it is a new move a Frame 1 to Frame 2 b Frame 2 to Frame 3 c Frame 3 to Frame 4 d Frame 4 to Frame 5 e Frame 5 to Frame 6 2 8 How would you describe the new 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 G8M1 LESSON 2 ACTIVITY 2 3 Your teacher will give you a set of cards Sort the cards into categories according to the type of move they show Be prepared to describe each category and why it is different from the others 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

G8M1 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary Here are the moves we have learned about so far A A translation slides a figure without turning it Every point in the figure goes the same distance in the same direction For example Figure A was translated down and to the left as shown by the arrows Figure B is a translation of Figure A A C 45 B A rotation turns a figure about a point called the center of the rotation Every point on the figure goes in a circle around the center and makes the same angle The rotation can be clockwise going in the same direction as the hands of a clock or counterclockwise going in the other direction For example Figure A was rotated 45 clockwise around its bottom vertex Figure C is a rotation of Figure A A reflection places points on the opposite side of a reflection line The mirror image is a backwards copy of the original figure The reflection line shows where the mirror should stand For example Figure A was reflected across the dotted line Figure D is a reflection of Figure A A D We use the word image to describe the new figure created by moving the original figure If one point on the original figure moves to another point on the new figure we call them corresponding points TERMINOLOGY Clockwise Corresponding Counterclockwise Image Reflection Rotation Translation 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 G8M1 LESSON 2 Name Date GRADE 8 MISSION 1 LESSON 2 Exit Ticket What type of move takes Figure A to Figure B A 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 11

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ZEARN MATH STUDENT EDITION G8M1 LESSON 3 Lesson 3 Grid Moves Let s transform some figures on grids Warm Up 1 What do you notice What do you wonder 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

G8M1 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Your teacher will give you tracing paper to carry out the moves specified Use A B and C to indicate vertices in the new figure that correspond to the points A B and C in the original figure A C A A B C B Figure 1 Figure 2 O C A Figure 3 C A B B 14 C Figure 4 1 In Figure 1 translate triangle ABC so that A goes to A 2 In Figure 2 translate triangle ABC so that C goes to C 3 In Figure 3 rotate triangle ABC 90 counterclockwise using center O 4 In Figure 4 reflect triangle ABC using line 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 G8M1 LESSON 3 Your teacher will give you tracing paper to carry out the moves specified Use A B C and D to indicate vertices in the new figure that correspond to the points A B C and D in the original figure B A B A C D C D Figure 5 Figure 6 B B A C D Figure 7 A C D Figure 8 1 In Figure 5 rotate quadrilateral ABCD 60 counterclockwise using center B 2 In Figure 6 rotate quadrilateral ABCD 60 clockwise using center C 3 In Figure 7 reflect quadrilateral ABCD using line 4 In Figure 8 translate quadrilateral ABCD so that A goes to C 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

G8M1 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary When a figure is on a grid we can use the grid to describe a transformation For example here is a figure and an image of the figure after a move Quadrilateral is translated ABCD 4 units to the right and 3 units down to the position of quadrilateral A B C D C D B A C D A B A second type of grid is called an isometric grid The isometric grid is made up of equilateral triangles The angles in the triangles all measure 60 degrees making the isometric grid convenient for showing rotations of 60 degrees Here is quadrilateral KLMN and its image K L M N after a 60 degree counterclockwise rotation around a point P M L N L K P K M N 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 G8M1 LESSON 3 Name Date GRADE 8 MISSION 1 LESSON 3 Exit Ticket Which of these triangles are translations of Triangle A Select all that apply A E D B C 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 17

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ZEARN MATH STUDENT EDITION G8M1 LESSON 4 Lesson 4 Making the Moves Let s draw and describe translations rotations and reflections Warm Up 1 Here is triangle ABD Your teacher will flash a completed image of triangle ABD twice Your job is to complete the image here A D B Concept Exploration ACTIVITY 1 2 Your partner will describe the image of this triangle after a certain transformation Sketch it here C B A 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

G8M1 LESSON 4 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Here are some figures on an isometric grid A B C 20 1 Name a transformation that takes Figure A to Figure B Name a transformation that takes Figure B to Figure C 2 What is one sequence of transformations that takes Figure A to Figure C 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 G8M1 LESSON 4 Lesson Summary A move or combination of moves is called a transformation When we do one or more moves in a row we often call that a sequence of transformations To distinguish the original figure from its image points in the image are sometimes labeled with the same letters as the original figure but with the symbol attached as in A pronounced A prime A translation can be described by two points If a translation moves point A to point A it moves the entire figure the same distance and direction as the distance and direction from A to A The distance and direction of a translation can be shown by an arrow For example here is a translation of quadrilateral ABCD that moves A to A D A B D A B C C A rotation can be described by an angle and a center The direction of the angle can be clockwise or counterclockwise For example hexagon ABCDEF is rotated 90 counterclockwise using center P P A F B C E D B C A D F E A reflection can be described by a line of reflection the mirror Each point is reflected directly across the line so that it is just as far from the mirror line but is on the opposite side For example pentagon ABCDE is reflected across line m m A B A D C E B D E C 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

G8M1 LESSON 4 ZEARN MATH STUDENT EDITION TERMINOLOGY Sequence of transformations Transformation 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 G8M1 LESSON 4 Name Date GRADE 8 MISSION 1 LESSON 4 Exit Ticket A B C 1 If you were to describe a translation of triangle ABC what information would you need to include in your description 2 If you were to describe a rotation of triangle ABC what information would you need to include in your description 3 If you were to describe a reflection of triangle ABC what information would you need to include in your description 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 G8M1 LESSON 5 Lesson 5 Coordinate Moves Let s transform some figures and see what happens to the coordinates of points Warm Up 1 Select all of the translations below that take Triangle T to Triangle U There may be more than one correct answer y 5 4 3 2 1 0 3 0 5 4 3 T 2 1 0 1 2 1 2 4 3 1 2 U 1 2 1 2 3 4 5 x 0 1 3 4 5 1 Translate 3 0 to 1 2 2 Translate 2 1 to 2 1 3 Translate 4 3 to 0 1 4 Translate 1 2 to 2 1 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

G8M1 LESSON 5 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Below is a list of points and a coordinate plane Follow the directions below y 10 8 6 4 2 0 10 8 6 4 2 0 2 2 4 6 8 10 x 4 6 8 10 Here is a list of points A 0 5 4 1 B 4 5 C 7 2 D 6 0 E 0 3 On the coordinate plane a Plot each point and label each with its coordinates b Using the x axis as the line of reflection plot the image of each point c Label the image of each point with its coordinates d Include a label using a letter For example the image of point A should be labeled A 2 26 If the point 13 10 were reflected using the x axis as the line of reflection what would be the coordinates of the image What about 13 20 13 570 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 3 G8M1 LESSON 5 The point R has coordinates 3 2 Answer the questions below y 5 4 3 R 2 1 0 5 4 3 2 1 0 1 1 2 3 4 5 x 2 3 4 5 a Without graphing predict the coordinates of the image of point R if point R were reflected using the y axis as the line of reflection b Check your answer by finding the image of R on the graph c Label the image of point R as R d What are the coordinates of R e Suppose you reflect a point using the y axis as the line of reflection How would you describe its image 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

G8M1 LESSON 5 ZEARN MATH STUDENT EDITION ACTIVITY 2 43 Apply each of the transformations listed below to segment AB shown below y 8 6 4 A 0 3 2 0 6 4 2 0 2 2 4 B 4 2 6 8 10 x 4 28 1 Rotate segment AB 90 degrees counterclockwise around center B Label the image of A as C What are the coordinates of C 2 Rotate segment AB 90 degrees counterclockwise around center A Label the image of B as D What are the coordinates of D 3 Rotate segment AB 90 degrees clockwise around 0 0 Label the image of A as E and the image of B as F What are the coordinates of E and F 4 Compare the two 90 degree counterclockwise rotations of segment AB What is the same about the images of these rotations What is 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

ZEARN MATH STUDENT EDITION G8M1 LESSON 5 Lesson Summary We can use coordinates to describe points and find patterns in the coordinates of transformed points We can describe a translation by expressing it as a sequence of horizontal and vertical translations For example segment AB is translated right 3 and down 2 y A 2 1 5 4 3 B 1 2 2 1 0 2 1 0 1 1 2 3 B 4 0 x 4 5 A 1 1 2 Reflecting a point across an axis changes the sign of one coordinate For example reflecting the point A whose coordinates are 2 1 across the x axis changes the sign of the y coordinate making its image the point A whose coordinates are 2 1 Reflecting the point A across the y axis changes the sign of the x coordinate making the image the point A whose coordinates are 2 1 y 2 A 2 1 1 0 5 4 3 2 A 2 1 1 0 1 1 2 3 4 5 x A 2 1 2 Reflections across other lines are more complex to describe We don t have the tools yet to describe rotations in terms of coordinates in general Here is an example of a 90 rotation with center 0 0 in a counterclockwise direction y B 3 2 2 1 0 5 4 3 B 2 3 3 2 1 0 1 A 0 0 1 2 3 4 5 x Point A has coordinates 0 0 Segment AB was rotated 90 counterclockwise around A Point B with coordinates 2 3 rotates to point B whose coordinates are 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 29

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ZEARN MATH STUDENT EDITION G8M1 LESSON 5 Name Date GRADE 8 MISSION 1 LESSON 5 Exit Ticket One of the triangles pictured is a rotation of triangle ABC and one of them is a reflection y 8 6 C 4 2 0 8 6 4 2 0 B A 2 4 6 8 x 2 4 6 8 1 Identify the center of rotation and label the rotated image PQR 2 Identify the line of reflection and label the reflected image XYZ 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 6 Lesson 6 Describing Transformations Let s transform some polygons in the coordinate plane Warm Up 1 Andre performs a 90 degree counterclockwise rotation of polygon P and gets polygon P but he does not say what the center of the rotation is Can you find the center P P 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

G8M1 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 You will receive 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 Before telling your partner the information ask Why do you need that information Solve the problem and explain your reasoning to your partner 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 34 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 G8M1 LESSON 6 Lesson Summary The center of a rotation for a figure doesn t have to be one of the points on the figure To find a center of rotation look for a point that is the same distance from two corresponding points You will probably have to do this for a couple of different pairs of corresponding points to nail it down When we perform a sequence of transformations the order of the transformations can be important Here is triangle ABC translated up two units and then reflected over the x axis Here is triangle ABC reflected over the x axis and then translated up two units Triangle ABC ends up in different places when the transformations are applied in the opposite order 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 Name G8M1 LESSON 6 Date GRADE 8 MISSION 1 LESSON 6 Exit Ticket Jada applies two transformations to a polygon in the coordinate plane One of the transformations is a translation and the other is a reflection What information does Jada need to provide to communicate the transformations she has used 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 7 Lesson 7 No Bending or Stretching Let s compare measurements before and after translations rotations and reflections Warm Up 1 1 Complete the problems below For each question the unit is represented by the large tick marks with whole numbers Find the length of this segment to the nearest 18 of a unit 1 2 4 5 6 2 3 4 5 Estimate the length of this segment to the nearest 18 of a unit 1 4 3 Find the length of this segment to the nearest 0 1 of a unit 1 3 2 2 3 4 5 Estimate the length of the segment in the prior question to the nearest 0 1 of a unit 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

G8M1 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Solve the problems below You can use tracing paper to help you draw the images of the figures or to check your work Translate Polygon A so point P goes to point Q In the image write the length of each side in grid units next to the side P Q A 2 Rotate Triangle B 90 degrees clockwise using R as the center of rotation In the image write the measure of each angle in the interior 30 B 90 60 3 R Reflect Pentagon C across line a In the image write the length of each side in grid units next to the side You may need to make your own ruler with tracing paper or a blank index card b In the image write the measure of each angle in the interior 90 150 C 90 40 90 120 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 G8M1 LESSON 7 ACTIVITY 2 3 Here is a grid showing triangle ABC and two other triangles D E B C F A G You can use a rigid transformation to take triangle ABC to one of the other triangles 1 Which one Explain how you know 2 Describe a rigid transformation that takes ABC to the triangle you selected 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

G8M1 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary The transformations we ve learned about so far translations rotations reflections and sequences of these motions are all examples of rigid transformations A rigid transformation is a move that doesn t change measurements on any figure Earlier we learned that a figure and its image have corresponding points With a rigid transformation figures like polygons also have corresponding sides and corresponding angles These corresponding parts have the same measurements For example triangle EFD was made by reflecting triangle ABC across a horizontal line then translating Corresponding sides have the same lengths and corresponding angles have the same measures B A 63 4 E 2 83 2 24 71 6 45 3 00 C 63 4 2 24 D 3 00 71 6 F Measurements in triangle ABC Corresponding measurements in image DEF AB 2 24 EF 2 24 BC 2 83 FD 2 83 CA 3 00 DE 3 00 m ABC 71 6 m EFD 71 6 m BCA 45 0 m FDE 45 0 m CAB 63 4 m DEF 63 4 45 2 83 B TERMINOLOGY Rigid transformation 42 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 G8M1 LESSON 7 Name Date GRADE 8 MISSION 1 LESSON 7 Exit Ticket Trapezoid A B C D is the image of trapezoid ABCD under a rigid transformation A D 130 50 B C 6 4 Label all vertices on trapezoid A B C D On both figures label all known side lengths and angle measures 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 8 Lesson 8 Rotation Patterns Let s rotate figures in a plane Warm Up 1 Here is a right isosceles triangle C B A 1 Rotate triangle ABC 90 degrees clockwise around B 2 Rotate triangle ABC 180 degrees clockwise around B 3 Rotate triangle ABC 270 degrees clockwise around B 4 What would it look like if you rotated the four triangles 90 degrees clockwise around B 180 degrees clockwise 270 degrees clockwise 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

G8M1 LESSON 8 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Solve the problems below using the grid E D G C 46 1 Rotate segment CD 180 degrees around point D Draw its image and label the image of C as A 2 Rotate segment CD 180 degrees around point E Draw its image and label the image of C as B and the image of D as F 3 Rotate segment CD 180 degrees around its midpoint G What is the image of C 4 What happens when you rotate a segment 180 degrees around a 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

ZEARN MATH STUDENT EDITION G8M1 LESSON 8 ACTIVITY 2 3 You can use rigid transformations of a figure to make patterns Here is a diagram built with three different transformations of triangle ABC C B D A E H G F 1 Describe a rigid transformation that takes triangle ABC to triangle CDE 2 Describe a rigid transformation that takes triangle ABC to triangle EFG 3 Describe a rigid transformation that takes triangle ABC to triangle GHA 4 Do segments AC CE EG and GA all have the same lengths 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 47

G8M1 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary When we apply a 180 degree rotation to a line segment there are several possible outcomes The segment maps to itself if the center of rotation is the midpoint of the segment The image of the segment overlaps with the segment and lies on the same line if the center of rotation is a point on the segment The image of the segment does not overlap with the segment if the center of rotation is not on the segment We can also build patterns by rotating a shape For example triangle ABC shown here has m A 60 If we rotate triangle ABC 60 degrees 120 degrees 180 degrees 240 degrees and 300 degrees clockwise we can build a hexagon B A 48 C 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 G8M1 LESSON 8 Name Date GRADE 8 MISSION 1 LESSON 8 Exit Ticket Here are two triangles A B Is triangle B the image of triangle A after a rotation 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 G8M1 LESSON 9 Lesson 9 Moves in Parallel Let s transform some lines Warm Up 1 For each diagram describe a translation rotation or reflection that takes line to line Then plot and label A and B the images of A and B 1 B A 2 B A 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

G8M1 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Use a piece of tracing paper to trace lines a and b and point K Then use that tracing paper to draw the images of the lines under the three different transformations listed in your notes As you perform each transformation think about the question What is the image of two parallel lines under a rigid transformation 1 Translate lines a and b 3 units up and 2 units to the right K a h b a What do you notice about the changes that occur to lines a and b after the translation b What is the same in the original and the image 2 Rotate lines a and b counterclockwise 180 degrees using K as the center of rotation K a h b a What do you notice about the changes that occur to lines a and b after the rotation b What is the same in the original and the image 52 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 G8M1 LESSON 9 Reflect lines a and b across line h K a h b a What do you notice about the changes that occur to lines a and b after the reflection b What is the same in the original and the image ACTIVITY 2 3 1 Answer the questions below The diagram shows a line with points labeled A B C and D a On the diagram draw the image of the line and points A C and B after the line has been rotated 180 degrees around point D b Label the images of the points A B and C c What is the order of all seven points Explain or show your reasoning A C D 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 53

G8M1 LESSON 9 2 ZEARN MATH STUDENT EDITION The diagram shows a line with points A and C on the line and a segment AD where D is not on the line a Rotate the figure 180 degrees about point C Label the image of A as A and the image of D as D b What do you know about the relationship between angle CAD and angle CA D Explain or show your reasoning D A C 3 The diagram shows two lines and m that intersect at a point O with point A on and point D on m a Rotate the figure 180 degrees around O Label the image of A as A and the image of D as D b What do you know about the relationship between the angles in the figure Explain or show your reasoning D A O m 54 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 G8M1 LESSON 9 Lesson Summary Rigid transformations have the following properties A rigid transformation of a line is a line A rigid transformation of two parallel lines results in two parallel lines that are the same distance apart as the original two lines Sometimes a rigid transformation takes a line to itself For example A B F B A m A translation parallel to the line The arrow shows a translation of line m that will take m to itself A rotation by 180 around any point on the line A 180 rotation of line m around point F will take m to itself A reflection across any line perpendicular to the line A reflection of line m across the dashed line will take m to itself These facts let us make an important conclusion If two lines intersect at a point which we ll call O then a 180 rotation of the lines with center O shows that vertical angles are congruent Here is an example 140 A 40 C C 40 140 A Rotating both lines by 180 around O sends angle AOC to angle A OC proving that they have the same measure The rotation also sends angle AOC to angle A OC TERMINOLOGY Vertical angles 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 9 Name Date GRADE 8 MISSION 1 LESSON 9 Exit Ticket Points A B and C are the images of 180 degree rotations of A B and C respectively around point O B 79 A C C 35 O A B Answer each question and explain your reasoning without measuring segments or angles 1 Name a segment whose length is the same as segment AO 2 What is the measure of angle A OB 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 G8M1 LESSON 10 Lesson 10 Composing Figures Let s use reasoning about rigid transformations to find measurements without measuring Warm Up 1 Here is triangle ABC Solve the problems below A 3 B 3 2 C 1 Reflect triangle ABC over line AB Label the image of C as C 2 Rotate triangle ABC around A so that C matches up with B 3 What can you say about the measures of angles B and C 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

G8M1 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here is triangle ABC Solve the problems below A B 60 C 1 Draw midpoint M of side AC 2 Rotate triangle ABC 180 degrees using center M to form triangle CDA Draw and label this triangle 3 What kind of quadrilateral is ABCD 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 G8M1 LESSON 10 ACTIVITY 2 3 The picture shows 3 triangles Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations A E 3 D 2 1 B C 1 Describe a rigid transformation that takes Triangle 1 to Triangle 2 What points in Triangle 2 correspond to points A B and C in the original triangle 2 Describe a rigid transformation that takes Triangle 1 to Triangle 3 What points in Triangle 3 correspond to points A B and C in the original triangle 3 Find two pairs of line segments in the diagram that are the same length and explain how you know they are the same length 4 Find two pairs of angles in the diagram that have the same measure and explain how you know they have the same measure 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

G8M1 LESSON 10 ZEARN MATH STUDENT EDITION Lesson Summary Earlier we learned that if we apply a sequence of rigid transformations to a figure then corresponding sides have equal lengths and corresponding angles have equal measures These facts let us figure things out without having to measure them For example here is triangle ABC C 36 B A We can reflect triangle ABC across side AC to form a new triangle B C 36 36 A B Because points A and C are on the line of reflection they do not move So the image of triangle ABC is B AC We also know that Angle B AC measures 36 because it is the image of angle BAC Segment AB has the same length as segment AB When we construct figures using copies of a figure made with rigid transformations we know that the measures of the images of segments and angles will be equal to the measures of the original segments and angles 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 G8M1 LESSON 10 Name Date GRADE 8 MISSION 1 LESSON 10 Exit Ticket Here is a diagram showing triangle ABC and some transformations of triangle ABC A D A 2 7 B E 2 7 64 3 B 64 3 M 3 2 3 2 C C On the left side of the diagram triangle ABC has been reflected across line AC to form quadrilateral ABCD On the right side of the diagram triangle ABC has been rotated 180 degrees using midpoint M as a center to form quadrilateral ABCE Using what you know about rigid transformations side lengths and angle measures label as many side lengths and angle measures as you can in quadrilaterals ABCD and ABCE 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 G8M1 LESSON 11 Lesson 11 What Is the Same Let s decide whether shapes are the same Warm Up 1 A person s hands are mirror images of each other In the diagram a left hand is labeled Shade all of the right hands Left hand Concept Exploration ACTIVITY 1 2 For each pair of shapes decide whether or not they are the same 1 2 3 4 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 65

G8M1 LESSON 11 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Answer the questions below 3 2 2 R 3 1 1 6 1 C 1 6 2 3 A 6 B 6 1 4 2 D 1 3 1 3 4 4 3 2 2 66 F 4 E 1 1 Which of these rectangles have the same area as Rectangle R but different perimeter 2 Which rectangles have the same perimeter as Rectangle R but different area 3 Which have the same area as Rectangle R and the same perimeter 4 Use materials from the geometry tool kit to decide which rectangles are congruent Shade congruent rectangles with the same color 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 G8M1 LESSON 11 Lesson Summary Congruent is a new term for an idea we have already been using We say that two figures are congruent if one can be lined up exactly with the other by a sequence of rigid transformations For example triangle EFD is congruent to triangle ABC because they can be matched up by reflecting triangle ABC across AC followed by the translation shown by the arrow Notice that all corresponding angles and side lengths are equal B 71 6 2 83 2 24 A 63 4 45 3 00 E C 3 00 63 4 45 2 24 D 2 83 71 6 F B Here are some other facts about congruent figures We don t need to check all the measurements to prove two figures are congruent we just have to find a sequence of rigid transformations that match up the figures A figure that looks like a mirror image of another figure can be congruent to it This means there must be a reflection in the sequence of transformations that matches up the figures Since two congruent polygons have the same area and the same perimeter one way to show that two polygons are not congruent is to show that they have a different perimeter or area TERMINOLOGY Congruent 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

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ZEARN MATH STUDENT EDITION Name G8M1 LESSON 11 Date GRADE 8 MISSION 1 LESSON 11 Exit Ticket Figure B is the image of Figure A when reflected across line l Are Figure A and Figure B congruent Explain your reasoning Figure A Figure 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 69

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ZEARN MATH STUDENT EDITION G8M1 LESSON 12 Lesson 12 Congruent Polygons Let s decide if two figures are congruent Warm Up 1 All of these triangles are congruent Sometimes we can take one figure to another with a translation Shade the triangles that are images of triangle ABC under a translation A B C 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

G8M1 LESSON 12 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 For each of the following pairs of shapes decide whether or not they are congruent Explain your reasoning 1 y B C D H A F x E G 2 y C E J I D H F B A 72 G 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 G8M1 LESSON 12 3 y F E C B D x A 4 y N M O L P K I J F E G D x H C A 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 73

G8M1 LESSON 12 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Are the two shapes congruent K I F E D A 43 C L J H G B For each pair of shapes decide whether or not Shape A is congruent to Shape B Explain how you know 1 y J H Q P B K I A R S L U T x G 74 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 G8M1 LESSON 12 2 y G H Q A L K J B R T S x I P 3 y H A I U G J R Q P B S 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 75

G8M1 LESSON 12 ZEARN MATH STUDENT EDITION 4 y S H G R B I A P Q x J 5 y Q J I P A G H B R S x 76 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 G8M1 LESSON 12 Lesson Summary How do we know if two figures are congruent If we copy one figure on tracing paper and move the paper so the copy covers the other figure exactly then that suggests they are congruent We can prove that two figures are congruent by describing a sequence of translations rotations and reflections that move one figure onto the other so they match up exactly How do we know that two figures are not congruent If there is no correspondence between the figures where the parts have equal measure that proves that the two figures are not congruent In particular If two polygons have different sets of side lengths they can t be congruent For example the figure on the left has side lengths 3 2 1 1 2 1 The figure on the right has side lengths 3 3 1 2 2 1 There is no way to make a correspondence between them where all corresponding sides have the same length If two polygons have the same side lengths but their orders can t be matched as you go around each polygon the polygons can t be congruent For example rectangle ABCD can t be congruent to quadrilateral EFGH Even though they both have two sides of length 3 and two sides of length 5 they don t correspond in the same order In ABCD the order is 3 5 3 5 or 5 3 5 3 in EFGH the order is 3 3 5 5 or 3 5 5 3 or 5 5 3 3 H 3 5 A 3 B 5 D E 3 3 C F 5 5 G If two polygons have the same side lengths in the same order but different corresponding angles the polygons can t be congruent For example parallelogram JKLM can t be congruent to rectangle ABCD Even though they have the same side lengths in the same order the angles are different All angles in ABCD are right angles In JKLM angles J and L are less than 90 degrees and angles K and M are more than 90 degrees 5 M L 3 J 3 5 K 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 12 Name Date GRADE 8 MISSION 1 LESSON 12 Exit Ticket Describe a sequence of reflections rotations and translations that shows that quadrilateral ABCD is congruent to quadrilateral EFGH A D E F H B C G 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 G8M1 LESSON 13 Lesson 13 Congruence Let s find ways to test congruence of interesting figures Warm Up 1 Trapezoids ABCD and A B C D are congruent A D A E B C F G D H B Draw and label the points on A B C D that correspond to E and F Draw and label the points on ABCD that correspond to G and H Draw and label at least three more pairs of corresponding points C Concept Exploration ACTIVITY 1 2 Are any of the ovals congruent to one another 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 81

G8M1 LESSON 13 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Here are two congruent shapes with some corresponding points labeled B A C E D C A 82 1 Draw the points corresponding to B D and E and label them B D and E 2 Draw line segments AD and A D and measure them Do the same for segments BC and B C and for segments AE and A E What do you notice 3 Do you think there could be a pair of corresponding segments with different lengths Explain 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 G8M1 LESSON 13 Lesson Summary To show two figures are congruent you align one with the other by a sequence of rigid transformations This is true even for figures with curved sides Distances between corresponding points on congruent figures are always equal even for curved shapes For example corresponding segments AB and A B on these congruent ovals have the same length A A B B To show two figures are not congruent you can find parts of the figures that should correspond but have different measurements For example these two ovals don t look congruent On both the longest distance is 5 units across and the longest distance from top to bottom is 4 units The line segment from the highest to lowest point is in the middle of the left oval but in the right oval it s 2 units from the right end and 3 units from the left end This proves they are not congruent 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 13 Name Date GRADE 8 MISSION 1 LESSON 13 Exit Ticket Are Figures A and B congruent Explain your reasoning Figure A Figure 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 85

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ZEARN MATH STUDENT EDITION G8M1 LESSON 14 Lesson 14 Alternate Interior Angles Let s explore why some angles are always equal Warm Up 1 Use the diagram below to complete the questions H J G 30 F I 1 Find the measure of angle JGH Explain or show your reasoning 2 Find and label a second 30 angle in the diagram Find and label an angle congruent to angle JGH 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

G8M1 LESSON 14 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Lines AC and DF are parallel They are cut by transversal HJ Use this diagram to answer the questions D H E A F 63 B C J 1 With your partner find the seven unknown angle measures in the diagram Explain your reasoning 2 What do you notice about the angles with vertex B and the angles with vertex E 3 Using what you noticed find the measures of the four angles at point B in the second diagram Lines AC and DF are parallel G D A B E 34 F C H 88 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 4 G8M1 LESSON 14 The next diagram resembles the first one but the lines form slightly different angles Work with your partner to find the six unknown angles with vertices at points B and E D H A E 108 63 B C J 5 F What do you notice about the angles in this diagram as compared to the earlier diagram How are the two diagrams different How are they the same ACTIVITY 2 3 Lines l and k are parallel and t is a transversal Point M is the midpoint of segment PQ Find a rigid transformation showing that angles MPA and MQB are congruent Q B M k P A 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 89

G8M1 LESSON 14 43 ZEARN MATH STUDENT EDITION In this picture lines l and k are no longer parallel M is the still the midpoint of segment PQ Does your argument in the earlier problem apply in this situation Explain Q B M k A P t 90 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 G8M1 LESSON 14 Lesson Summary When two lines intersect vertical angles are equal and adjacent angles are supplementary that is their measures sum to 180 For example in this figure angles 1 and 3 are equal angles 2 and 4 are equal angles 1 and 4 are supplementary and angles 2 and 3 are supplementary 70 1 110 2 4 110 3 70 When two parallel lines are cut by another line called a transversal two pairs of alternate interior angles are created Interior means on the inside or between the two parallel lines For example in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles 70 1 110 2 4 110 3 70 70 5 110 6 8 110 7 70 Alternate interior angles are equal because a 180 rotation around the midpoint of the segment that joins their vertices takes each angle to the other Imagine a point M halfway between the two intersections can you see how rotating 180 about M takes angle 3 to angle 5 Using what we know about vertical angles adjacent angles and alternate interior angles we can find the measures of any of the eight angles created by a transversal if we know just one of them For example starting with the fact that angle 1 is 70 we use vertical angles to see that angle 3 is 70 then we use alternate interior angles to see that angle 5 is 70 then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is 110 since 180 70 110 It turns out that there are only two different measures In this example angles 1 3 5 and 7 measure 70 and angles 2 4 6 and 8 measure 110 TERMINOLOGY Alternate Interior Angles Transversal 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 91

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ZEARN MATH STUDENT EDITION G8M1 LESSON 14 Name Date GRADE 8 MISSION 1 LESSON 14 Exit Ticket The diagram shows two parallel lines cut by a transversal One angle measure is shown e d a 54 f g b c Find the values of a b c d e f and g 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 15 Lesson 15 Adding the Angles in a Triangle Let s angles in triangles Warm Up 1 Complete the questions below 1 Complete the table by drawing a triangle in each cell that has the properties listed for its column and row If you think you cannot draw a triangle with those properties write impossible in the cell 2 Share your drawings with a partner Discuss your thinking If you disagree work to reach an agreement Acute all angles acute Right has a right angle Obtuse has an obtuse angle Scalene side lengths all different Isosceles at least two side lengths are equal Equilateral three side lengths equal 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

G8M1 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Your teacher will give you a card with a picture of a triangle 1 The measurement of one of the angles is labeled Mentally estimate the measures of the other two angles 2 Find two other students with triangles congruent to yours but with a different angle labeled Confirm that the triangles are congruent that each card has a different angle labeled and that the angle measures make sense 3 Enter the three angle measures for your triangle on the table your teacher has posted ACTIVITY 2 3 96 Your teacher will give you a page with three sets of angles and a blank space Cut out each set of three angles Can you make a triangle from each set that has these same three angles 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 G8M1 LESSON 15 Lesson Summary A 180 angle is called a straight angle because when it is made with two rays they point in opposite directions and form a straight line If we experiment with angles in a triangle we find that the sum of the measures of the three angles in each triangle is 180 the same as a straight angle Through experimentation we find If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides then the three angles form a straight angle If we have a line and two rays that form three angles added to make a straight angle then there is a triangle with these three angles TERMINOLOGY Straight angle 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 Name G8M1 LESSON 15 Date GRADE 8 MISSION 1 LESSON 15 Exit Ticket In triangle ABC the measure of angle B is 50 degrees 1 Give possible values for the measures of angles A and C if ABC is an acute triangle 2 Give possible values for the measures of angles A and C if ABC is an obtuse triangle 3 Give possible values for the measures of angles A and C if ABC is a right triangle 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 16 Lesson 16 Parallel Lines and the Angles in a Triangle Let s see why the angles in a triangle add to 180 degrees Warm Up 1 Is each equation true or false 1 62 28 60 30 2 3 8 2 8 8 3 16 24 40 2 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 101

G8M1 LESSON 16 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here is triangle ABC Answer the questions below A B 102 C 1 Rotate triangle ABC 180 around the midpoint of side AC Label the new vertex D 2 Rotate triangle ABC 180 around the midpoint of side AB Label the new vertex E 3 Look at angles EAB BAC and CAD Without measuring write what you think is the sum of the measures of these angles Explain or show your reasoning 4 Is the measure of angle EAB equal to the measure of any angle in triangle ABC If so which one If not how do you know 5 Is the measure of angle CAD equal to the measure of any angle in triangle ABC If so which one If not how do you know 6 What is the sum of the measures of angles ABC BAC and ACB 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 G8M1 LESSON 16 ACTIVITY 2 3 Here is triangle ABC Line DE is parallel to line AC D B E b A a c C 1 What is m DBA b m CBE Explain how you know 2 Use your answer to explain why a b c 180 3 Explain why your argument will work for any triangle that is explain why the sum of the angle measures in any triangle is 180 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

G8M1 LESSON 16 ZEARN MATH STUDENT EDITION ACTIVITY 3 43 1 For each figure below determine the missing angle measures Note that the figures may not be drawn to scale Line l is parallel to line k l k 115 40 2 70 60 3 30 50 104 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 G8M1 LESSON 16 Lesson Summary Using parallel lines and rotations we can understand why the angles in a triangle always add to 180 Here is triangle ABC Line DE is parallel to AC and contains B E D B A 180 degree rotation of triangle ABC around the z x midpoint of AB interchanges angles A and DBA y so they have the same measure in the picture these angles are marked as x A 180 degree rotation of triangle ABC around the midpoint of BC interchanges angles C and CBE so they have the same measure in the picture these angles are x z marked as z Also DBE is a straight line because A C 180 degree rotations take lines to parallel lines So the three angles with vertex B make a line and they add up to 180 x y z 180 But x y z are the measures of the three angles in ABC so the sum of the angles in a triangle is always 180 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

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ZEARN MATH STUDENT EDITION Name G8M1 LESSON 16 Date GRADE 8 MISSION 1 LESSON 16 Exit Ticket 1 In an equilateral triangle all side lengths are equal and all angle measures are equal Sketch an equilateral triangle What are the measures of its angles 2 In an isosceles triangle which is not equilateral two side lengths are equal and two angle measures are equal Sketch three different isosceles triangles 3 List two different possibilities for the angle measures of an isosceles triangle 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

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ZEARN MATH STUDENT EDITION G8M1 LESSON 17 Lesson 17 Rotate and Tessellate Let s make complex patterns using transformations Warm Up 1 Your teacher will give you some shapes Use them to answer the questions below 1 How many copies of the equilateral triangle can you fit together around a single vertex so that the triangles edges have no gaps or overlaps What is the measure of each angle in these triangles 2 What are the measures of the angles in the a square b hexagon c parallelogram d right triangle e octagon f pentagon 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

G8M1 LESSON 17 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 110 Solve the problems below 1 Design your own tessellation You will need to decide which shapes you want to use and make copies Remember that a tessellation is a repeating pattern that goes on forever to fill up the entire plane 2 Find a partner and trade pictures Describe a transformation of your partner s picture that takes the pattern to itself How many different transformations can you find that take the pattern to itself Consider translations reflections and rotations 3 If there s time color and decorate your tessellation 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 G8M1 LESSON 17 ACTIVITY 2 3 Solve the problems below 1 Make a design with rotational symmetry 2 Find a partner who has also made a design Exchange designs and find a transformation of your partner s design that takes it to itself Consider rotations reflections and translations 3 If there s time color and decorate your design 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

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Grade 8 Mission 2 Dilations Similarity and Introducing Slope

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ZEARN MATH STUDENT EDITION G8M2 LESSON 1 Lesson 1 Projecting and Scaling Let s explore scaling Warm Up 1 Find each quotient Write your answer as a fraction or a mixed number 1 6 14 2 2 10 17 5 3 8 12 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 115

G8M2 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 2 Rectangles were made by cutting an 8 12 inch by 11 inch piece of paper in half in half again and so on as illustrated in the diagram Find the lengths of each rectangle and enter them in the appropriate table Then solve problems 3 5 Some of the rectangles are scaled copies of the full sheet of paper Rectangle A Enter the measurements of those rectangles in the table Rectangle Length of short side inches Length of long side inches A 8 12 11 Some of the rectangles are not scaled copies of the full sheet of paper Enter the measurements of those rectangles in the table Rectangle 3 116 Length of short side inches Length of long side inches Look at the measurements for the rectangles that are scaled copies of the full sheet of paper What do you notice about the measurements of these rectangles Look at the measurements for the rectangles that are not scaled copies of the full sheet What do you notice about these 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

ZEARN MATH STUDENT EDITION G8M2 LESSON 1 4 Stack the rectangles that are scaled copies of the full sheet so that they all line up at a corner as shown in the diagram Do the same with the other set of rectangles On each stack draw a line from the bottom left corner to the top right corner of the biggest rectangle What do you notice 5 Stack all of the rectangles from largest to smallest so that they all line up at a corner Compare the lines that you drew Can you tell from the drawn lines which set each rectangle came from 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

G8M2 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary Scaled copies of rectangles have an interesting property Can you see what it is 6 4 3 2 Here the larger rectangle is a scaled copy of the smaller one with a scale factor of 32 Notice how the diagonal of the large rectangle contains the diagonal of the smaller rectangle This is the case for any two scaled copies of a rectangle if we line them up as shown If two rectangles are not scaled copies of one another then the diagonals do not match up In this mission we will investigate how to make scaled copies of a figure 118 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 G8M2 LESSON 1 Date GRADE 8 MISSION 2 LESSON 1 Exit Ticket In your own words explain what a dilation is 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 2 Lesson 2 Circular Grid Let s dilate figures on circular grids Warm Up 1 What do you notice What do you wonder 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

G8M2 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 The larger circle circle d is a dilation of the smaller circle circle c P is the center of dilation d c P 1 Draw four points on the smaller circle not inside the circle and label them E F G and H 2 Draw the rays from P through each of those four points 3 Label the points where the rays meet the larger circle E F G and H 4 Complete the table In the row labeled S write the distance between P and the point on the smaller circle in grid units In the row labeled L write the distance between P and the corresponding point on the larger circle in grid units E F G H S L 5 122 The center of dilation is point P What is the scale factor that takes the smaller circle to the larger circle 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 G8M2 LESSON 2 ACTIVITY 2 3 Here is polygon ABCD C B P A D 1 Dilate each vertex of polygon ABCD using P as the center of dilation and a scale factor of 2 Label the image of A as A and label the images of the remaining three vertices as B C and D 2 Draw segments between the dilated points to create polygon A B C D 3 What are some things you notice about the new polygon 4 Choose a few more points on the sides of the original polygon and transform them using the same dilation What do you notice 5 Dilate each vertex of polygon ABCD using P as the center of dilation and a scale factor of 12 Label the image of A as E the image of B as F the image of C as G and the image of D as H 6 What do you notice about polygon EFGH 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

G8M2 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary A circular grid like this one can be helpful for performing dilations The radius of the smallest circle is one unit and the radius of each successive circle is one unit more than the previous one To perform a dilation we need a center of dilation a scale factor and a point to dilate In the picture P is the center of dilation With a scale factor of 2 each point stays on the same ray from P but its distance from P doubles A A B B P C C Since the circles on the grid are the same distance apart segment PA has twice the length of segment PA and the same holds for the other points TERMINOLOGY Center of dilation Dilation 124 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 G8M2 LESSON 2 Name Date GRADE 8 MISSION 2 LESSON 2 Exit Ticket A P B 1 Dilate A using P as the center of dilation and a scale factor of 3 Label the new point A 2 Dilate B using P as the center of dilation and a scale factor of 2 Label the new point 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 125

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ZEARN MATH STUDENT EDITION G8M2 LESSON 3 Lesson 3 Dilations with no Grid Let s dilate figures not on grids Warm Up 1 Use the ray below to perform the dilations B A 1 Find and label a point C on the ray whose distance from A is twice the distance from B to A 2 Find and label a point D on the ray whose distance from A is half the distance from B to A 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

G8M2 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here is a diagram that shows nine points C A B D E F G H I 128 1 Dilate B using a scale factor of 5 and A as the center of dilation Which point is its image 2 Using H as the center of dilation dilate G so that its image is E What scale factor did you use 3 Using H as the center of dilation dilate E so that its image is G What scale factor did you use 4 To dilate F so that its image is B what point on the diagram can you use as a center 5 Dilate H using A as the center and a scale factor of 13 Which point is its image 6 Describe a dilation that uses a labeled point as its center and that would take F to H 7 Using B as the center of dilation dilate H so that its image is itself What scale factor did you use 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 G8M2 LESSON 3 ACTIVITY 2 3 Follow the directions below to perform the dilations P C Q 1 Using one colored pencil draw the images of points P and Q using C as the center of dilation and a scale factor of 4 Label the new points P and Q 2 Using a different color draw the images of points P and Q using C as the center of dilation and a scale factor of 12 Label the new points P and Q Pause here so your teacher can review your diagram Your teacher will then give you a scale factor to use in the next part 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

G8M2 LESSON 3 43 ZEARN MATH STUDENT EDITION Now you ll make a perspective drawing Here is a rectangle a Choose a point inside the shaded circular region but outside the rectangle to use as the center of dilation Label it C b Using your center C and the scale factor you were given draw the image under the dilation of each vertex of the rectangle one at a time Connect the dilated vertices to create the dilated rectangle c Draw a segment that connects each of the original vertices with its image This will make your diagram look like a cool three dimensional drawing of a box If there s time you can shade the sides of the box to make it look more realistic d Compare your drawing to other people s drawings What is the same and what is different How do the choices you made affect the final drawing Was your dilated rectangle closer to C than to the original rectangle or farther away How is that decided 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 G8M2 LESSON 3 Lesson Summary If A is the center of dilation how can we find which point is the dilation of B with scale factor 2 C B D A Since the scale factor is larger than 1 the point must be farther away from A than B is which makes C the point we are looking for If we measure the distance between A and C we would find that it is exactly twice the distance between A and B A dilation with scale factor less than 1 brings points closer Point D is the dilation of B with center A and scale factor 13 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 G8M2 LESSON 3 Name Date GRADE 8 MISSION 2 LESSON 3 Exit Ticket Lin drew a triangle and a dilation of the triangle with scale factor 12 C D B A E 1 What is the center of the dilation Explain how you know 2 Which triangle is the original and which triangle is the dilation 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 4 Lesson 4 Dilations on a Square Grid Let s dilate figures on a rectangular grid Warm Up 1 Point C is the dilation of point B with center of dilation A and scale factor s Estimate s Be prepared to explain your reasoning A B C Concept Exploration ACTIVITY 1 2 Find the dilation of quadrilateral ABCD with center P and scale factor 2 A B P D C 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

G8M2 LESSON 4 3 ZEARN MATH STUDENT EDITION Use the diagram below to dilate triangle QRS using the given scale factors a Find the dilation of triangle QRS with center T and scale factor 2 b Find the dilation of triangle QRS with center T and scale factor 12 Q T R S ACTIVITY 2 43 You will receive some cards Follow the directions below Each of Cards 1 through 6 shows a figure in the coordinate plane and describes a dilation Each of Cards A through E describes the image of the dilation for one of the numbered cards Match the number cards with the letter cards One of the number cards will not have a match For this card you ll need to draw an image 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 G8M2 LESSON 4 Lesson Summary Square grids can be useful for showing dilations The grid is helpful especially when the center of dilation and the point s being dilated lie at grid points Rather than using a ruler to measure the distance between the points we can count grid units For example suppose we want to dilate point Q with center of dilation P and scale factor 32 Since Q is 4 grid squares to the left and 2 grid squares down from P the dilation will be 6 grid squares to the left and 3 grid squares down from P can you see why The dilated image is marked as Q in the picture P Q Q Sometimes the square grid comes with coordinates The coordinate grid gives us a convenient way to name points and sometimes the coordinates of the image can be found with just arithmetic For example to make a dilation with center 0 0 and scale factor 2 of the triangle with coordinates 1 2 3 1 and 2 1 we can just double the coordinates to get 2 4 6 2 and 4 2 y 5 4 3 6 2 2 3 1 1 7 6 5 4 3 2 1 2 2 4 1 1 2 3 1 2 3 4 5 6 7 x 2 1 4 2 4 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 137

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ZEARN MATH STUDENT EDITION G8M2 LESSON 4 Name Date GRADE 8 MISSION 2 LESSON 4 Exit Ticket Draw the image of rectangle ABCD under dilation using center P and scale factor 21 B C P A 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 139

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ZEARN MATH STUDENT EDITION G8M2 LESSON 5 Lesson 5 More Dilations Let s look at dilations in the coordinate plane Warm Up 1 All of the triangles are dilations of triangle D The dilations use the same center P but different scale factors What do triangles A B and C have in common What do triangles E F and G have in common What does this tell us about the different scale factors used P A B C D E F G Concept Exploration ACTIVITY 1 2 You will receive 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 2 2 3 4 Silently read your card and think about what information you need to answer the question Ask your partner for the specific information that you need Explain to your partner how you are using the information to solve the problem Solve the problem and explain your reasoning to your partner 3 4 Silently read the information on your card 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 Before telling your partner the information ask Why do you need that information After your partner solves the problem ask them to explain their reasoning and listen to their explanation 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 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

G8M2 LESSON 5 ZEARN MATH STUDENT EDITION Lesson Summary One important use of coordinates is to communicate geometric information precisely Let s consider a quadrilateral ABCD in the coordinate plane Performing a dilation of ABCD requires three vital pieces of information 1 The coordinates of A B C and D 2 The coordinates of the center of dilation P 3 The scale factor of the dilation With this information we can dilate the vertices A B C and D and then draw the corresponding segments to find the dilation of ABCD Without coordinates describing the location of the new points would likely require sharing a picture of the polygon and the center of dilation 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 G8M2 LESSON 5 Name Date GRADE 8 MISSION 2 LESSON 5 Exit Ticket The smaller triangle is dilated to create the larger triangle The center of dilation is plotted but not labeled y 4 3 2 1 2 1 1 1 2 3 4 5 6 7 8 9 10 11 x 2 3 4 5 6 7 Describe this dilation Be sure to include all of the information someone would need to perform the dilation 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 G8M2 LESSON 6 Lesson 6 Similarity Let s explore similar figures Warm Up 1 Use what you know about operations and their properties to write three expressions equivalent to the expression shown 10 2 3 8 3 Concept Exploration ACTIVITY 1 2 Triangle EGH and triangle LME are similar Find a sequence of translations rotations reflections and dilations that shows this H G E M L 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

G8M2 LESSON 6 Hexagon ABCDEF and hexagon HGLKJI are similar Find a sequence of translations rotations reflections and dilations that shows this 3 A ZEARN MATH STUDENT EDITION G B C F H D E K L J I ACTIVITY 2 Sketch figures similar to Figure A that use only the transformations listed to show similarity 43 A 146 1 A translation and a reflection Label your sketch Figure B Pause here so that your teacher can check your work 2 A reflection and a dilation with scale factor greater than 1 Label your sketch Figure C 3 A rotation and a reflection Label your sketch Figure D 4 A dilation with scale factor less than 1 and a translation Label your sketch Figure E 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 G8M2 LESSON 6 Lesson Summary Let s show that triangle ABC is similar to triangle DEF E A B D F C Two figures are similar if one figure can be transformed into the other by a sequence of translations rotations reflections and dilations There are many correct sequences of transformations but we only need to describe one to show that two figures are similar One way to get from ABC to DEF follows these steps step 1 reflect across line f step 2 rotate 90 counterclockwise around D step 3 dilate with center D and scale factor 2 E B A D step 2 C F step 1 B C C B Another way would be to dilate triangle ABC by a scale factor of 2 with center of dilation A then translate A to D then reflect over a vertical line through D and finally rotate it so it matches up with triangle DEF What steps would you choose to show the two triangles are similar TERMINOLOGY Similar 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 6 Name Date GRADE 8 MISSION 2 LESSON 6 Exit Ticket Elena gives the following sequence of transformations to show that the two figures are similar by transforming ABCD into EFGD G C F B A 1 Dilate using center D and scale factor 2 2 Reflect using the line AE D E Is Elena s method correct If not explain how you could fix it 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 G8M2 LESSON 7 Lesson 7 Similar Polygons Let s look at sides and angles of similar polygons Warm Up 1 Choose whether each of the statements is true in all cases in some cases or in no cases 1 If two figures are congruent then they are similar 2 If two figures are similar then they are congruent 3 If an angle is dilated with the center of dilation at its vertex the angle measure may change Concept Exploration ACTIVITY 1 2 Let s look at a square and a rhombus 2 2 120 2 2 2 2 120 60 2 60 2 Priya says These polygons are similar because their side lengths are all the same Clare says These polygons are not similar because the angles are different Do you agree with either Priya or Clare 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 151

G8M2 LESSON 7 3 ZEARN MATH STUDENT EDITION Now let s look at rectangles ABCD and EFGH H D 4 2 A G C 2 4 6 4 4 B E 6 F Jada says These rectangles are similar because all of the side lengths differ by 2 Lin says These rectangles are similar I can dilate AD and BC using a scale factor of 2 and AB and CD using a scale factor of 1 5 to make the rectangles congruent Then I can use a translation to line up the rectangles Do you agree with either Jada or Lin Explain your reasoning ACTIVITY 2 43 Your teacher will give you a card Find someone else in the room who has a card with a polygon that is similar but not congruent to yours When you have found your partner work with them to explain how you know that the two polygons are similar 152 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 G8M2 LESSON 7 Lesson Summary When two polygons are similar Every angle and side in one polygon has a corresponding part in the other polygon All pairs of corresponding angles have the same measure Corresponding sides are related by a single scale factor Each side length in one figure is multiplied by the scale factor to get the corresponding side length in the other figure Consider the two rectangles shown here Are they similar A E B 4 F 3 2 3 H D G C It looks like rectangles ABCD and EFGH could be similar if you match the long edges and match the short edges All the corresponding angles are congruent because they are all right angles Calculating the scale factor between the sides is where we see that looks like isn t enough to make them similar To scale the long side AB to the long side EF the scale factor must be 34 because 4 34 3 But the scale 2 factor to match AD to EH has to be 23 because 3 3 2 So the rectangles are not similar because the scale factors for all the parts must be the same Here is an example that shows how sides can correspond with a scale factor of 1 but the quadrilaterals are not similar because the angles don t have the same measure 3 2 3 2 3 2 2 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 153

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ZEARN MATH STUDENT EDITION G8M2 LESSON 7 Name Date GRADE 8 MISSION 2 LESSON 7 Exit Ticket Explain how you know these two figures are similar C D 4 125 55 4 5 6 6 E 55 A 125 4 B G 125 3 55 55 125 3 F H 4 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 155

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ZEARN MATH STUDENT EDITION G8M2 LESSON 8 Lesson 8 Similar Triangles Let s look at similar triangles Warm Up 1 Create three different expressions that are each equal to 20 Each expression should include only these three numbers 4 2 and 10 Concept Exploration ACTIVITY 1 2 Create a triangle using three pieces of pasta and angle A Your triangle must include the angle you were given but you are otherwise free to make any triangle you like Tape your pasta triangle to a sheet of paper so it won t move a After you have created your triangle measure each side length with a ruler and record the length on the paper next to the side Then measure the angles to the nearest 5 degrees using a protractor and record these measurements on your paper b Find two others in the room who have the same angle A and compare your triangles What is the same What is different Are the triangles congruent Similar c How did you decide if they were or were not congruent or similar 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

G8M2 LESSON 8 3 ZEARN MATH STUDENT EDITION Now use more pasta and angles A B and C to create another triangle Tape this pasta triangle on a separate sheet of paper a After you have created your triangle measure each side length with a ruler and record the length on the paper next to the side Then measure the angles to the nearest 5 degrees using a protractor and record these measurements on your paper b Find two others in the room who used your same angles and compare your triangles What is the same What is different Are the triangles congruent Similar c How did you decide if they were or were not congruent or similar 43 Here is triangle PQR Break a new piece of pasta different in length than segment PQ R P Q Tape the piece of pasta so that it lays on top of line PQ with one end of the pasta at P if it does not fit on the page break it further Label the other end of the piece of pasta S Tape a full piece of pasta with one end at S making an angle congruent to PQR Tape a full piece of pasta on top of line PR with one end of the pasta at P Call the point where the two full pieces of pasta meet T a Is your new pasta triangle PST similar to PQR Explain your reasoning b If your broken piece of pasta were a different length would the pasta triangle still be similar to PQR Explain your reasoning 158 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 G8M2 LESSON 8 Lesson Summary We learned earlier that two polygons are similar when there is a sequence of translations rotations reflections and dilations taking one polygon to the other When the polygons are triangles we only need to check that both triangles have two corresponding angles to show they are similar can you tell why Here is an example Triangle ABC and triangle DEF each have a 30 degree angle and a 45 degree angle D 30 E A 45 30 F 45 C B We can translate A to D and then rotate so that the two 30 degree angles are aligned giving this picture A D 30 B E C 45 F Now a dilation with center D and appropriate scale factor will move C to F This dilation also moves B to E showing that triangles ABC and DEF are similar 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 8 Name Date GRADE 8 MISSION 2 LESSON 8 Exit Ticket Here are two triangles y y 6 6 5 5 C 4 4 3 3 2 2 1 1 A 1 2 3 B C A 4 5 x B 1 1 Show that the triangles are similar 2 What is the scale factor from triangle ABC to triangle A B C 2 3 4 5 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 161

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ZEARN MATH STUDENT EDITION G8M2 LESSON 9 Lesson 9 Side Length Quotients in Similar Triangles Let s find missing side lengths in triangles Warm Up 1 Triangle A has side lengths 2 3 and 4 Triangle B has side lengths 4 5 and 6 Is Triangle A similar to Triangle B How do you know Concept Exploration ACTIVITY 1 2 Triangle ABC is similar to triangles DEF GHI and JKL Complete the first table You will be assigned one of the 3 columns in the second table The scale factors for the dilations that show triangle ABC is similar to each triangle are in the first table C 7 5 B 4 A 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

G8M2 LESSON 9 ZEARN MATH STUDENT EDITION Triangle Scale factor Length of short side Length of medium side Length of long side ABC 1 4 5 7 DEF 2 GHI 3 JKL 1 2 Triangle long side short side ABC 7 4 or 1 75 long side medium side medium side short side DEF GHI JKL 164 1 Find the side lengths of triangles DEF GHI and JKL Record them in the first table 2 For all four triangles find the quotient of the triangle side lengths assigned to you and record them in the second table What do you notice about the quotients 3 Compare your results with your partner s and complete your 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

ZEARN MATH STUDENT EDITION G8M2 LESSON 9 ACTIVITY 2 3 Triangles ABC EFD and GHI are all similar The side lengths of the triangles all have the same units Find the unknown side lengths 5 F E C 4 4 e c B d A I 12 5 H 6 5 h G 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 165

G8M2 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary If two polygons are similar then the side lengths in one polygon are multiplied by the same scale factor to give the corresponding side lengths in the other polygon For these triangles the scale factor is 2 C 4 3 A 8 6 C 5 B A B 10 Here is a table that shows relationships between the short and medium length sides of the small and large triangle Small triangle Large triangle Medium side 4 8 Short side 3 6 Medium side Short side 4 3 8 4 6 3 The lengths of the medium side and the short side are in a ratio of 4 3 This means that the medium side in each triangle is 43 as long as the short side This is true for all similar polygons the ratio between two sides in one polygon is the same as the ratio of the corresponding sides in a similar polygon We can use these facts to calculate missing lengths in similar polygons For example triangles A B C and ABC shown here are similar Let s find the length of segment B C 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 G8M2 LESSON 9 Name Date GRADE 8 MISSION 2 LESSON 9 Exit Ticket The two triangles shown are similar Find the value of ab a b 2 1 1 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 167

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ZEARN MATH STUDENT EDITION G8M2 LESSON 10 Lesson 10 Meet Slope Let s learn about the slope of a line Warm Up 1 Write some numbers that are equal to 15 12 Concept Exploration ACTIVITY 1 2 1 The figure shows three right triangles each with its longest side on the same line Your teacher will assign you two triangles Explain why the two triangles are similar E H F 2 G C 6 D 3 A 2 4 B Complete the table Triangle Vertical side Horizontal side vertical side horizontal side ABC CDE FGH 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

G8M2 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 170 Use your straightedge to draw lines with a given slope 1 Draw two lines with slope 3 What do you notice about the two lines 2 Draw two lines with slope 12 What do you notice about the two 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

ZEARN MATH STUDENT EDITION G8M2 LESSON 10 Lesson Summary Here is a line drawn on a grid There are also four right triangles drawn Do you notice anything the triangles have in common A 6 2 3 3 C D 1 2 B 4 E 4 6 These four triangles are all examples of slope triangles One side of a slope triangle is on the line one side is vertical and another side is horizontal The slope of the line is the quotient of the length of the vertical side and the length of the horizontal side of the slope triangle This number is the same for all slope triangles for the same line because all slope triangles for the same line are similar In this example the slope of the line is 32 which is what all four triangles have in common Here is how the slope is calculated using the slope triangles Points A and B give 2 3 23 Points D and B give 4 6 23 Points A and C give 4 6 23 Points A and E give 23 1 23 TERMINOLOGY Slope 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 10 Name Date GRADE 8 MISSION 2 LESSON 10 Exit Ticket Lines and k are graphed k 1 Which line has a slope of 1 and which has a slope of 2 2 Use a ruler to help you graph a line whose slope is 13 Label this line a 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 G8M2 LESSON 11 Lesson 11 Writing Equations for Lines Let s explore the relationship between points on a line and the slope of the line Warm Up 1 Find each of the following and explain your reasoning y D 4 7 E A 0 2 B 2 2 C 4 2 x 1 The length of segment BE 2 The coordinates of E 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

G8M2 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Line j is shown in the coordinate plane Answer the questions about line j below y 10 9 8 j 7 D 6 5 4 B 3 2 1 A 1 1 176 C 1 2 3 4 5 6 7 8 9 10 x 1 What are the coordinates of B and D 2 Is point 20 15 on line j Explain how you know 3 Is point 100 75 on line j Explain how you know 4 Is point 90 68 on line j Explain how you know 5 Suppose you know the x and y coordinates of a point Write a rule that would allow you to test whether the point is on line j 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 G8M2 LESSON 11 ACTIVITY 2 Here are two diagrams Answer the questions about the diagrams below 3 y y 12 10 9 8 6 A 4 3 1 D 0 1 1 6 C 4 E x y 5 4 D 2 4 F 3 2 F 2 A 7 C 4 3 8 E x y 2 1 9 3 5 B 10 B 7 1 11 k 1 3 4 5 6 7 8 9 10 x 1 1 1 2 3 4 5 6 7 8 9 10 11 12 x 1 Complete each diagram so that all vertical and horizontal segments have expressions for their lengths 2 Use what you know about similar triangles to find an equation for the quotient of the vertical and horizontal side lengths of DFE in each 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 177

G8M2 LESSON 11 ZEARN MATH STUDENT EDITION Lesson Summary y Here are the points A C and E on the same line Triangles ABC and ADE are slope triangles for the line so we know they are similar triangles Let s use their similarity to better understand the relationship between x and y which make up the coordinates of point E 8 6 5 The slope for triangle ABC is 21 since the vertical side 4 has length 2 and the horizontal side has length 1 The slope we find for triangle ADE is yx because the vertical side has length y and the horizontal side has length x These two slopes must be equal since they are from slope triangles for the same line and so 2 y 1 x 2 A B 1 1 1 D x 0 2 3 4 5 6 x 6 x x y E x y 8 7 The slope for triangle ABC is 21 since the vertical side has length 2 and the horizontal side has length 1 For triangle ADE the horizontal side has length x The vertical side has length y 1 because the distance from x y to the x axis is y but the vertical side of the triangle stops 1 unit short of the x axis So the slope we find for triangle ADE is y x 1 The slopes for the two slope triangles are equal meaning 178 C 1 Here are two different slope triangles We can use the same reasoning to describe the relationship between x and y for this point E Since y 1 is twice x another way to write this equation is y 1 2x This equation is true for any point x y on the line y 3 Since 21 2 this means that the value of y is twice the value of x or that y 2x This equation is true for any point x y on the line 2 y 1 1 x E x y 7 6 5 y 1 4 3 C 2 A B 1 1 1 D x 1 x 2 3 4 5 1 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 G8M2 LESSON 11 Name Date GRADE 8 MISSION 2 LESSON 11 Exit Ticket y 13 12 11 10 9 x y 8 a 5 7 7 6 5 4 3 2 1 2 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 x 2 1 Explain why the slope of line a is 26 2 Label the horizontal and vertical sides of the triangle with expressions representing their length 3 Explain why yx 57 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 179

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ZEARN MATH STUDENT EDITION G8M2 LESSON 12 Lesson 12 Using Equations for Lines Let s write equations for lines Warm Up A dilation with scale factor 2 sends A to B Where is the center of the dilation 1 B A Concept Exploration ACTIVITY 1 Here is a line Answer the questions about this line below 2 y 10 1 Using what you know about similar triangles find an equation for the line in the diagram 2 What is the slope of this line Does it appear in your equation 3 Is 9 11 also on the line How do you know 4 Is 100 193 also on the line x y 9 8 7 7 7 6 5 4 5 3 3 2 1 1 2 3 4 5 6 7 8 9 10 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 181

G8M2 LESSON 12 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Here is triangle ABC Answer the questions about this triangle below 5 4 3 C 2 1 A B 1 182 2 3 4 5 6 7 8 1 Draw the dilation of triangle ABC with center 0 1 and scale factor 2 2 Draw the dilation of triangle ABC with center 0 1 and scale factor 2 5 3 Where is C mapped by the dilation with center 0 1 and scale factor s 4 For which scale factor does the dilation with center 0 1 send C to 9 5 5 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 G8M2 LESSON 12 Lesson Summary We can use what we know about slope to decide if a point lies on a line Here is a line with a few points labeled y 6 5 4 2 5 x y 3 2 1 0 1 1 2 3 4 5 6 x The slope triangle with vertices 0 1 and 2 5 gives a slope of 52 10 2 The slope triangle with vertices 0 1 and x y gives a slope of y x 1 Since these slopes are the same y x 1 2 is an equation for the line So if we want to check whether or not the point 11 23 lies on this line we can check that 2311 1 2 Since 11 23 is a solution to the equation it is on the line 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

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ZEARN MATH STUDENT EDITION G8M2 LESSON 12 Name Date GRADE 8 MISSION 2 LESSON 12 Exit Ticket y 10 9 8 7 6 5 4 3 2 1 1 1 1 2 3 4 5 6 7 8 9 10 x Is the point 20 13 on this line 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 185

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ZEARN MATH STUDENT EDITION G8M2 LESSON 13 Lesson 13 The Shadow Knows Let s use shadows to find the heights of an object Warm Up 1 What do you notice What do you wonder Whole Group Lesson ACTIVITY 1 2 Use the photo and the table below to explore the relationship between an object s height and the length of its shadow 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

G8M2 LESSON 13 ZEARN MATH STUDENT EDITION Here are some measurements that were taken when the photo was taken It was impossible to directly measure the height of the lamppost so that cell is blank Height inches Shadow length inches Younger boy 43 29 Man 72 48 Older boy 51 34 Lamppost 114 1 What relationships do you notice between an object s height and the length of its shadow 2 Make a conjecture about the height of the lamppost and explain your thinking ACTIVITY 2 3 188 Explain why the relationship between the height of these objects and the length of their shadows is approximately proportional 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 G8M2 LESSON 13 ACTIVITY 3 43 Let s apply what you have learned about shadow lengths of different objects to estimate the height of an object outside 1 Head outside Make sure that it is a sunny day and you take a measuring device like a tape measure or meter stick as well as a pencil and some paper 2 Choose an object whose height is too large to measure directly Your teacher may assign you an object 3 Use what you have learned to figure out the height of the object 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 189

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ZEARN MATH STUDENT EDITION G8M3 LESSON 1 Lesson 1 Understanding Proportional Relationships Let s study some graphs Warm Up 1 What do you notice What do you wonder 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

G8M3 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 A ladybug and ant move at constant speeds The diagrams with tick marks show their positions at different times Each tick mark represents 1 centimeter start start start start 194 1 Lines u and v also show the positions of the two bugs Which line shows the ladybug s movement Which line shows the ant s movement Explain your reasoning 2 How long does it take the ladybug to travel 12 cm The ant 3 Scale the vertical and horizontal axes by labeling each grid line with a number You will need to use the time and distance information shown in the tick mark diagrams 4 Mark and label the point on line u and the point on line v that represent the time and position of each bug after traveling 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

ZEARN MATH STUDENT EDITION G8M3 LESSON 1 ACTIVITY 2 3 Use the tick mark diagrams and graph in the earlier activity when needed to solve the problems below 1 Imagine a bug that is moving twice as fast as the ladybug On each tick mark diagram mark the position of this bug 2 Plot this bug s positions on the coordinate axes with lines u and v and connect them with a line 3 Write an equation for each of the three 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 195

G8M3 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary Graphing is a way to help us make sense of relationships But the graph of a line on a coordinate axes without a scale or labels isn t very helpful For example let s say we know that on longer bike rides Kiran can ride 4 miles every 16 minutes and Mai can ride 4 miles every 12 minutes Here are the graphs of these relationships Without labels we can t even tell which line is Kiran and which is Mai Without labels and a scale on the axes we can t use these graphs to answer questions like 1 Which graph goes with which rider 2 Who rides faster 3 If Kiran and Mai start a bike trip at the same time how far are they after 24 minutes 4 How long will it take each of them to reach the end of the 12 mile bike path Here are the same graphs but now with labels and scale 48 10 40 time minutes 40 32 10 30 24 4 16 16 8 4 12 1 4 1 3 2 4 6 8 10 12 14 distance miles 196 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 G8M3 LESSON 1 Revisiting the questions from earlier 1 Which graph goes with each rider If Kiran rides 4 miles in 16 minutes then the point 4 16 is on his graph If he rides for 1 mile it will take 4 minutes 10 miles will take 40 minutes So the upper graph represents Kiran s ride Mai s points for the same distances are 1 3 4 12 and 10 30 so hers is the lower graph A letter next to each line would help us remember which is which 2 Who rides faster Mai rides faster because she can ride the same distance as Kiran in a shorter time 3 If Kiran and Mai start a bike trip at the same time how far are they after 20 minutes The points on the graphs at height 20 are 5 miles for Kiran and a little less than 7 miles for Mai 4 How long will it take each of them to reach the end of the 12 mile bike path The points on the graphs at a horizontal distance of 12 are 36 minutes for Mai and 48 minutes for Kiran Kiran s time after 12 miles is almost off the grid 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 Name G8M3 LESSON 1 Date GRADE 8 MISSION 3 LESSON 1 Exit Ticket This graph represents the positions of two turtles in a race 1 On the same axes draw a line for a third turtle that is going half as fast as the turtle described by line g 2 Explain how your line shows that the turtle is going half as fast 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 G8M3 LESSON 2 Lesson 2 Graphs of Proportional Relationships Let s think about scale Warm Up Here is a graph that could represent a variety of different situations 1 1 Write an equation for the graph y 2 Sketch a new graph of this relationship y 28 210 26 175 24 22 140 20 105 18 16 70 14 35 12 10 20 8 40 60 80 100 120 x 6 4 2 2 4 6 8 10 12 14 16 18 20 22 x Concept Exploration ACTIVITY 1 2 Your teacher will give you 12 graphs of proportional relationships 1 Sort the graphs into groups based on what proportional relationship they represent 2 Write an equation for each different proportional relationship you find 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

G8M3 LESSON 2 ZEARN MATH STUDENT EDITION ACTIVITY 2 Tank A is filling with water The two graphs below show the relationship between the volume of water and amount of time passed 3 Tank A is not filled at a constant rate and the relationship between its volume of water and time is graphed on each set of axes Tank B is filled at a constant rate of 12 liters per minute The relationship between its volume of water and time can be described by the equation v 21 t where t is the time in minutes and v is the total volume in liters of water in the tank v v 1 8 50 1 4 volume liters volume liters 1 6 1 2 1 0 8 0 6 0 4 40 30 20 10 0 2 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 t time minutes 20 40 60 time minutes 80 t 1 Sketch and label a graph of the relationship between the volume of water v and time t for Tank B on each of the axes 2 Answer the following questions and say which graph you used to find your answer a After 30 seconds which tank has the most water b At approximately what times do both tanks have the same amount of water c 202 At approximately what times do both tanks contain 1 liter of water 20 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

ZEARN MATH STUDENT EDITION G8M3 LESSON 2 Lesson Summary The scales we choose when graphing a relationship often depend on what information we want to know For example say two water tanks are filled at different constant rates The relationship between time in minutes t and volume in liters v of tank A is given by v 2 2t For tank B the relationship is v 2 75t If we want to use graphs to see at what times the two tanks will have 110 liters of water then using an axis scale from 0 to 10 as shown here isn t very helpful If we use a vertical scale that goes to 150 liters a bit beyond the 110 we are looking for and a horizontal scale that goes to 100 minutes we get a much more useful set of axes for answering our question v 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 time minutes 8 9 t Now we can see that the two tanks will reach 110 liters 10 minutes apart tank B after 40 minutes of filling and tank A after 50 minutes of filling 140 130 120 110 volume liters v volume liters These equations tell us that tank A is being filled at a constant rate of 2 2 liters per minute and tank B is being filled at a constant rate of 2 75 liters per minute It is important to note that both of these graphs are correct but one uses a range of values that helps answer the question In order to always pick a helpful scale we should consider the situation and the questions asked about it 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 t time minutes 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 G8M3 LESSON 2 Name Date GRADE 8 MISSION 3 LESSON 2 Exit Ticket Which one of these relationships is different than the other three Explain how you know A B y 6 4 2 0 2 0 4 0 6 0 8 C y 70 60 50 40 30 20 10 2 4 6 8 10 12 14 x x D y 40 y 60 30 40 20 20 10 2 4 6 8 x 20 40 60 80 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 205

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ZEARN MATH STUDENT EDITION G8M3 LESSON 3 Lesson 3 Representing Proportional Relationships Let s graph proportional relationships Warm Up 1 Find the value of each product mentally 15 2 15 0 5 15 0 25 15 2 25 Concept Exploration ACTIVITY 1 2 Here are two ways to represent Jada and Noah s steps Answer the questions about this situation below Description Jada and Noah counted the number of steps they took to walk a set distance To walk the same distance Jada took 8 steps Noah took 10 steps Equation Let x represent the number of steps Jada takes and let y represent the number of steps Noah takes y 54 x Then they found that when Noah took 15 steps Jada took 12 steps a Create a table that represents this situation with at least 3 pairs of values 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

G8M3 LESSON 3 ZEARN MATH STUDENT EDITION b Graph this relationship and label the axes c How can you see or calculate the constant of proportionality in each representation What does it mean d Explain how you can tell that the equation description graph and table all represent the same situation 3 Here are two ways to represent a fundraiser by the Origami Club Answer the questions about this situation below Description The Origami Club is doing a car wash fundraiser to raise money for a trip They charge the same price for every car After 11 cars they raised a total of 93 50 After 23 cars they raised a total of 195 50 Table Number of cars Amount raised in dollars 11 93 50 23 195 50 a Write an equation that represents this situation Use c to represent number of cars and use m to represent amount raised in dollars 208 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 G8M3 LESSON 3 b Create a graph that represents this situation m c c How can you see or calculate the constant of proportionality in each representation What does it mean d Explain how you can tell that the equation description graph and table all represent the same situation ACTIVITY 2 43 You will receive either a problem card or a data card Do not show or read your card to your partner Follow the directions in your notes based on the card you ve received 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 3 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 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 4 Solve the problem and explain your reasoning to your partner 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 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

G8M3 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary Proportional relationships can be represented in multiple ways Which representation we choose depends on the purpose And when we create representations we can choose helpful values by paying attention to the context For example a stew recipe calls for 3 carrots for every 2 potatoes One way to represent this is using an equation If there are p potatoes and c carrots then c 32 p Suppose we want to make a large batch of this recipe for a family gathering using 150 potatoes To find the number of carrots we could just use the equation 23 150 225 carrots Now suppose the recipe is used in a restaurant that makes the stew in large batches of different sizes depending on how busy a day it is using up to 300 potatoes at time Then we might make a graph to show how many carrots are needed for different amounts of potatoes We set up a pair of coordinate axes with a scale from 0 to 300 along the horizontal axis and 0 to 450 on the vertical axis because 450 32 300 Then we can read how many carrots are needed for any number of potatoes up to 300 Or if the recipe is used in a food factory that produces very large quantities and the potatoes come in bags of 150 we might just make a table of values showing the number of carrots needed for different multiplies of 150 Number of carrots 450 150 225 400 300 450 350 450 675 600 900 No matter the representation or the scale used the constant of proportionality 32 is evident in each In the equation it is the number we multiply by p in the graph it is the slope and in the table it is the number we multiply values in the left column to get numbers in the right column We can think of the constant of proportionality as a rate of change of c with respect to p In this case the rate of change is 32 carrots per potato Number of carrots Number of potatoes 300 250 200 150 100 50 50 100 150 200 250 Number of potatoes 300 TERMINOLOGY Rate of change 210 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 G8M3 LESSON 3 Name Date GRADE 8 MISSION 3 LESSON 3 Exit Ticket Sketch a graph that shows the relationship between grams of honey and grams of salt needed for a bakery recipe Show on the graph how much honey is needed for 70 grams of salt Salt g Honey g Flour c 10 14 4 25 35 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 211

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ZEARN MATH STUDENT EDITION G8M3 LESSON 4 Lesson 4 Comparing Proportional Relationships Let s compare proportional relationships Warm Up 1 The equation y 4 2x could represent a variety of different situations 1 Write a description of a situation represented by this equation Decide what quantities x and y represent in your situation 2 Make a table and a graph that represent 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 213

G8M3 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Elena and Jada each make money by helping out their neighbors Elena babysits Her earnings are given by the equation y 8 40x where x represents how many hours she works and y represents how much money she earns Jada earns 7 per hour mowing her neighbors lawns a Who makes more money after working 12 hours How much more do they make Explain how you know b What is the rate of change for each situation and what does it mean c 214 How long would it take each person to earn 150 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

ZEARN MATH STUDENT EDITION 3 G8M3 LESSON 4 Han and Clare have summer jobs stuffing envelopes for two different companies Han earns 15 for every 300 envelopes he finishes Clare s earnings Number of envelopes Money in dollars 400 40 900 90 a Who would make more money after stuffing 1 500 envelopes How much more money would they make Explain how you know b What is the rate of change for each situation and what does it mean c Who gets paid more in their job 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 215

G8M3 LESSON 4 43 ZEARN MATH STUDENT EDITION Tyler plans to start a lemonade stand and is trying out different recipes for lemonade He wants to make sure the recipe doesn t use too much lemonade mix lemon juice and sugar but still tastes good Recipe 1 is given by the equation y 4x where x represents the cups of lemonade mix and y represents the cups of water Recipe 2 Lemonade mix cups Water cups 10 50 13 65 21 105 a If Tyler had 16 cups of lemonade mix how many cups of water would he need for each recipe Explain how you know b What is the rate of change for each situation and what does it mean c 216 Tyler has a 5 gallon jug which holds 80 cups to use for his lemonade stand and 16 cups of lemonade mix Which lemonade recipe should he use 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

ZEARN MATH STUDENT EDITION G8M3 LESSON 4 Lesson Summary When two proportional relationships are represented in different ways we compare them by finding a common piece of information For example Clare s earnings are represented by the equation y 14 5x where y is the amount of money she earns in dollars for working x hours The table shows some information about Jada s pay Time worked hours Earnings dollars 10 50 13 65 21 105 Who is paid at a higher rate per hour How much more does that person have after 20 hours In Clare s equation we see that the constant of proportionality relating her earnings to time worked is 14 50 This means that she earns 14 50 per hour We can calculate Jada s constant of proportionality by dividing a value in the earnings column by a value in the same row in the time worked column Using the last row the constant of proportionality for Jada is 13 25 since 490 25 37 13 25 An equation representing Jada s earnings is y 13 25x This means she earns 13 25 per hour So Clare is paid at a higher rate than Jada Clare earns 1 25 more per hour than Jada which means that after 20 hours of work she has 20 1 25 25 more than Jada 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

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ZEARN MATH STUDENT EDITION G8M3 LESSON 4 Name Date GRADE 8 MISSION 3 LESSON 4 Exit Ticket Here are recipes for two mixtures of salt and water that taste different Mixture A Salt teaspoons Water cups 4 5 7 8 34 9 11 14 Mixture B is defined by the equation y 2 5x where x is the number of teaspoons of salt and y is the number of cups of water 1 If you used 10 cups of water which mixture would use more salt How much more Explain or show your reasoning 2 Which mixture tastes saltier 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 219

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ZEARN MATH STUDENT EDITION G8M3 LESSON 5 Lesson 5 Introduction to Linear Relationships Let s explore some relationships between two variables Warm Up 1 Find the value of 258 12 Concept Exploration ACTIVITY 1 2 We have two stacks of styrofoam cups How many cups are needed for a stack with a height of 50 cm 30 cm 29 cm 28 cm 27 cm 26 cm 25 cm 24 cm 23 cm 22 cm 21 cm 20 cm 19 cm 18 cm 17 cm 16 cm 15 cm 14 cm 13 cm 12 cm 11 cm 10 cm 9 cm 8 cm 7 cm 6 cm 5 cm 4 cm 3 cm 2 cm 1 cm 0 cm Stack of 6 cups height of 15 cm Stack of 12 cups height of 22 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 221

G8M3 LESSON 5 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Let s explore the structure of the stack of cups using a graph Height in centimeters 50 40 30 20 10 5 222 10 15 20 25 30 35 40 Number of cups 1 If you didn t create your own graph of the situation before do so now 2 What are some ways you can tell that the number of cups is not proportional to the height of the stack 3 What is the slope of the line in your graph What does the slope mean in this situation 4 At what point does your line intersect the vertical axis What do the coordinates of this point tell you about the cups 5 How much height does each cup after the first add to the stack 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 G8M3 LESSON 5 Lesson Summary Andre starts babysitting and charges 10 for traveling to and from the job and 15 per hour For every additional hour he works he charges another 15 If we graph Andre s earnings based on how long he works we have a line that starts at 10 on the vertical axis and then increases by 15 each hour A linear relationship is any relationship between two quantities where one quantity has a constant rate of change with respect to the other 160 Amount earned dollars 140 120 100 80 60 40 20 1 2 3 4 5 6 Time hours 7 8 9 We can figure out the rate of change using the graph Because the rate of change is constant we can take any two points on the graph and divide the amount of vertical change by the amount of horizontal change For example take the points 2 40 and 6 100 They mean that Andre earns 40 for working 2 40 hours and 100 for working 6 hours The rate of change is 100 6 2 15 dollars per hour Andre s earnings go up 15 for each hour of babysitting Notice that this is the same way we calculate the slope of the line That s why the graph is a line and why we call this a linear relationship The rate of change of a linear relationship is the same as the slope of its graph With proportional relationships we are used to graphs that contain the point 0 0 But proportional relationships are just one type of linear relationship In the following lessons we will continue to explore the other type of linear relationship where the quantities are not both 0 at the same time TERMINOLOGY Linear relationship Rate of change Slope 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

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ZEARN MATH STUDENT EDITION G8M3 LESSON 5 Name Date GRADE 8 MISSION 3 LESSON 5 Exit Ticket A shorter style of cup is stacked tall The graph displays the height of the stack in centimeters for different numbers of cups How much does each cup after the first add to the height of the stack Explain how you know 18 Height in centimeters 16 14 12 10 8 8 8 6 3 5 5 4 2 2 4 6 8 10 Number of cups 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 G8M3 LESSON 6 Lesson 6 More Linear Relationships Let s explore some more relationships between two variables Warm Up 1 Look for a growing pattern Describe the pattern you see 1 2 3 1 If your pattern continues growing in the same way how many tiles of each color will be in the 4th and 5th diagram The 10th diagram 2 How many tiles of each color will be in the nth diagram Be prepared to explain how your reasoning Concept Exploration ACTIVITY 1 2 1 Your teacher will give you 6 cards describing different situations and 6 cards with graphs Match each situation to a 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 227

G8M3 LESSON 6 2 ZEARN MATH STUDENT EDITION Pick one proportional relationship and one non proportional relationship and answer the following questions about them a How can you find the slope from the graph Explain or show your reasoning b Explain what the slope means in the situation c Find the point where the line crosses the vertical axis What does that point tell you about the situation ACTIVITY 2 Lin has a summer reading assignment After reading the first 30 pages of the book she plans to read 40 pages each day until she finishes Lin makes the graph shown here to track how many total pages she ll read over the next few days After day 1 Lin reaches page 70 which matches the point 1 70 she made on her graph After day 4 Lin reaches page 190 which does not match the point 4 160 she made on her graph Lin is not sure what went wrong since she knows she followed her reading plan 1 Sketch a line showing Lin s original plan on the axes y 160 140 Number of pages read 3 120 100 80 60 40 20 228 2 What does the vertical intercept mean in this situation How do the vertical intercepts of the two lines compare 3 What does the slope mean in this situation How do the slopes of the two lines compare 1 2 3 4 Number of days 5 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 G8M3 LESSON 6 Lesson Summary At the start of summer break Jada and Lin decide to save some of the money they earn helping out their neighbors to use during the school year Jada starts by putting 20 into a savings jar in her room and plans to save 10 a week Lin starts by putting 10 into a savings jar in her room plans to save 20 a week Here are graphs of how much money they will save after 10 weeks if they each follow their plans y 100 Amount saved dollars The value where a line intersects the vertical axis is called the vertical intercept When the vertical axis is labeled with a variable like y this value is also often called the y intercept Jada s graph has a vertical intercept of 20 while Lin s graph has a vertical intercept of 10 These values reflect the amount of money they each started with At 1 week they will have saved the same amount 30 But after week 1 Lin is saving more money per week so she has a larger rate of change so she will end up saving more money over the summer if they each follow their plans Lin 80 Jada 60 40 20 1 2 3 4 5 6 7 8 Time weeks 9 10 11 12 x TERMINOLOGY Vertical intercept 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

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ZEARN MATH STUDENT EDITION G8M3 LESSON 6 Name Date GRADE 8 MISSION 3 LESSON 6 Exit Ticket Savings in dollars The graph shows the savings in Andre s bank account 80 60 40 20 1 2 3 4 5 6 7 Time in weeks 1 What is the slope of the line 2 What is the meaning of the slope in this situation 8 9 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 231

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ZEARN MATH STUDENT EDITION G8M3 LESSON 7 Lesson 7 Representations of Linear Relationships Let s write equations from real situations Warm Up 1 Which glass will hold the most water The least 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

G8M3 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Answer the questions about the volume and number of objects Record data from your teacher s demonstration or your experiment in the table You may not need all the rows Number of objects Volume in ml 2 What is the volume V in the cylinder after you add x objects Explain your reasoning 3 If you wanted to make the water reach the highest mark on the cylinder how many objects would you need 4 Plot and label points that show your measurements from the experiment 120 Volume in ml V 100 80 60 40 20 2 4 6 8 10 12 Number of objects x 234 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 G8M3 LESSON 7 5 The points should fall on a line Use a ruler to graph this line 6 Compute the slope of the line What does the slope mean in this situation 7 What is the vertical intercept What does the vertical intercept mean in this situation ACTIVITY 2 3 Answer the questions about the graphs below y A y B y 4 9 C 30 75 10 3 5 x 1 x x For each graph record Vertical change 2 10 35 2 1 5 1 3 Horizontal change Slope Describe a procedure for finding the slope between any two points on a line 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 235

G8M3 LESSON 7 3 ZEARN MATH STUDENT EDITION Write an expression for the slope of the line in the graph using the letters u v s and t y u v s t x 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 G8M3 LESSON 7 Lesson Summary Let s say we have a glass cylinder filled with 50 ml of water and a bunch of marbles that are 3 ml in volume If we drop marbles into the cylinder one at a time we can watch the height of the water increase by the same amount 3 ml for each one added This constant rate of change means there is a linear relationship between the number of marbles and the height of the water Add one marble the water height goes up 3 ml Add 2 marbles the water height goes up 6 ml Add x marbles the water height goes up 3x ml Reasoning this way we can calculate that the height y of the water for x marbles is y 3x 50 Any linear relationship can be expressed in the form y mx b using just the rate of change m and the initial amount b The 3 represents the rate of change or slope of the graph and the 50 represents the initial amount or vertical intercept of the graph We ll learn about some more ways to think about this equation in future lessons Now what if we didn t have a description to use to figure out the slope and the vertical intercept That s okay so long as we can find some points on the line For the line graphed here two of the points on the line are 3 3 and 9 5 and we can use these points to draw in a slope triangle as shown y x The slope of this line is the quotient of the length of the vertical side of the slope triangle and the length vertical change 2 1 of the horizontal side of the slope triangle So the slope m is horizontal change 6 3 We can also see from the graph that the vertical intercept b is 2 Putting these together we can say that the equation for this line is y 31 x 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 237

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ZEARN MATH STUDENT EDITION Name G8M3 LESSON 7 Date GRADE 8 MISSION 3 LESSON 7 Exit Ticket Make a sketch of a linear relationship with a slope of 3 that is not a proportional relationship Show how you know that the slope is 3 Write an equation for the line 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

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ZEARN MATH STUDENT EDITION G8M3 LESSON 8 Lesson 8 Translating to y mx b Let s see what happens to the equations of translated lines Warm Up 1 The diagram shows several lines You can only see part of the lines but they actually continue forever in both directions j h g f i 1 Which lines are images of line f under a translation 2 For each line that is a translation of f draw an arrow on the grid that shows the vertical translation distance 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

G8M3 LESSON 8 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Answer the questions below using the coordinate plane Money saved dollars 2 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 11 Time babysitting hours 242 1 Diego earns 10 per hour babysitting Assume that he has no money saved before he starts babysitting and plans to save all of his earnings Graph how much money y he has after x hours of babysitting 2 Now imagine that Diego started with 30 saved before he starts babysitting On the same set of axes graph how much money y he would have after x hours of babysitting 3 Compare the second line with the first line How much more money does Diego have after 1 hour of babysitting 2 hours 5 hours x hours 4 Write an equation for each line 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 G8M3 LESSON 8 ACTIVITY 2 3 This graph shows two lines Line a goes through the origin 0 0 Line h is the image of line a under a translation y 5 4 3 2 a 1 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 3 4 5 1 x h Select all of the equations whose graph is the line h a y 14 x 5 b y 14 x 5 c d e f 43 1x 5 y 4 y 5 14 x 5 14 x y y 5 14 x You will each get 12 cards There are 4 pairs of lines A D showing the graph a of a proportional relationship and the image h of a under a translation Match each line h with an equation and either a table or description For the line with no matching equation write one on the blank card 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

G8M3 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary During an early winter storm the snow fell at a rate of 12 inches per hour We can see the rate of change 12 in both the equation that represents this storm y 12 x and in the slope of the line representing this storm In addition to being a linear relationship between the time since the beginning of the storm and the depth of the snow we can also call this a proportional relationship since the depth of snow was 0 at the beginning of the storm During a mid winter storm the snow again fell at a rate of 12 inches per hour but this time there was already 5 inches of snow on the ground We can graph this storm on the same axes as the first storm by taking all the points on the graph of the first storm and translating them up 5 inches 2 hours after each storm begins 1 inch of new snow has fallen For the first storm this means there is now 1 inch of snow on the ground For the second storm this means there are now 6 inches of snow on the ground Unlike the first storm the second is not a proportional relationship since the line representing the second storm has a vertical intercept of 5 The equation representing the storm y 12 x 5 is of the form y mx b M represents the rate of change and the slope of the graph and b represents the initial amount and the vertical intercept of the graph 244 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 G8M3 LESSON 8 Date GRADE 8 MISSION 3 LESSON 8 Exit Ticket Describe how the graph of y 2x is the same and different from the graph of y 2x 7 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 245

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ZEARN MATH STUDENT EDITION G8M3 LESSON 9 Lesson 9 Slopes Don t Have to be Positive Let s find out what a negative slope means Warm Up 1 Which line doesn t belong s v t u 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

G8M3 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Noah put 40 on his fare card Every time he rides public transportation 2 50 is subtracted from the amount available on his card How much money in dollars is available on his card after he takes a 0 rides b 1 ride c 2 rides d x rides 2 Graph the relationship between amount of money on the card and number of rides 45 dollars on card 40 35 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 number of rides 3 248 How many rides can Noah take before the card runs out of money Where do you see this number of rides on your 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 G8M3 LESSON 9 ACTIVITY 2 3 Here is a graph that shows the amount on Han s fare card for every day of last July 45 40 dollars on card 35 30 25 20 15 10 5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 days passed 1 Describe what happened with the amount on Han s fare card in July 2 Plot and label 3 different points on the line 3 Write an equation that represents the amount on the card in July y after x days 4 What value makes sense for the slope of the line that represents the amounts on Han s fare card in July 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

G8M3 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary depth of snow inches At the end of winter in Maine the snow on the ground was 30 inches deep Then there was a particularly warm day and the snow melted at the rate of 1 inch per hour The graph shows the relationship between the time since the snow started to melt and the depth of the snow 30 1 25 20 1 5 5 15 10 5 1 2 3 4 5 6 7 8 9 10 11 time since snow started to melt hours The slope of the graph is 1 since the rate of change is 1 inch per hour That is the depth goes down 1 inch per hour The vertical intercept is 30 since the snow was 30 inches deep when the warmth started to melt the snow The two slope triangles show how the rate of change is constant It just also happens to be negative in this case since after each hour that passes there is 1 inch less snow Graphs with negative slope often describe situations where some quantity is decreasing over time like the depth of snow on warm days or the amount of money on a fare card being used to take rides on buses Slopes can be positive negative or even zero A slope of 0 means there is no change in the y value even though the x value may be changing For example Elena won a contest where the prize was a special pass that gives her free bus rides for a year Her fare card had 5 on it when she won the prize Here is a graph of the amount of money on her fare card after winning the prize 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 G8M3 LESSON 9 balance on fare card dollars 6 2 days 5 5 5 days 5 4 3 2 1 1 2 3 4 5 6 7 8 time since winning contest days 9 10 11 The vertical intercept is 5 since the graph starts when she has 5 on her fare card The slope of the graph is 0 since she doesn t use her fare card for the next year meaning the amount on her fare card doesn t change for a year In fact all graphs of linear relationships with slopes equal to 0 are horizontal a rate of change of 0 means that from one point to the next the y values remain 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 251

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ZEARN MATH STUDENT EDITION G8M3 LESSON 9 Name Date GRADE 8 MISSION 3 LESSON 9 Exit Ticket Each square on a grid represents 1 unit on each side 1 Calculate the slope of graph D Explain or show your reasoning 2 Calculate the slope of graph E What situation could the graph represent 3 On the blank grid draw a line that passes through the indicated point and has slope 2 D E 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 253

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ZEARN MATH STUDENT EDITION G8M3 LESSON 10 Lesson 10 Calculating Slope Let s calculate slope from two points Warm Up 1 Find values for a and b that make each side have the same value 1 a b 2 2 a b 2 3 a b 2 Concept Exploration ACTIVITY 1 2 1 Answer the questions below about a line that passes through the points 1 11 and 8 2 Plot the points 1 11 and 8 2 and use a ruler to draw the line that passes through them y 11 10 2 Without calculating do you expect the slope of the line through 1 11 and 8 2 to be positive or negative How can you tell 9 8 7 6 5 3 Calculate the slope of this line 4 3 2 1 1 2 3 4 5 6 7 8 9 10 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 x 255

G8M3 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Your teacher will give you either a design or a blank graph Do not show your card to your partner If your teacher gives you the design If your teacher gives you the blank graph 1 1 Listen carefully as your partner describes each line and draw each line based on their description 2 You are not allowed to ask for more information about a line than what your partner tells you 3 Do not show your drawing to your partner until you have finished drawing all the lines they describe 2 3 Look at the design silently and think about how you could communicate what your partner should draw Think about ways that you can describe what a line looks like such as its slope or points that it goes through Describe each line one at a time and give your partner time to draw them Once your partner thinks they have drawn all the lines you described only then should you show them the design When finished place the drawing next to the card with the design so that you and your partner can both see them How is the drawing the same as the design How is it different Discuss any miscommunication that might have caused the drawing to look different from the design Pause here so your teacher can review your work When your teacher gives you a new set of cards switch roles for the second problem 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 G8M3 LESSON 10 Lesson Summary We learned earlier that one way to find the slope of a line is by drawing a slope triangle For example using the slope triangle shown here the slope of the line is 24 or 12 we know the slope is negative because the line is decreasing from left to right y 7 6 A 5 4 3 2 2 B 4 C 1 1 2 3 4 5 6 7 x But slope triangles are only one way to calculate the slope of a line Let s compute the slope of this line a different way using just the points A 1 5 and B 5 3 Since we know the slope is the vertical change divided by the horizontal change we can calculate the change in the y values and then the change in the x values Between points A and B the y value change is 3 5 2 and the x value change is 5 1 4 This means the slope is 24 or 12 which is the same as what we found using the slope triangle Notice that in each of the calculations we subtracted the value from point A from the value from point B If we had done it the other way around then the y value change would have been 5 3 2 and the x value change would have been 1 5 4 which still gives us a slope of 12 But what if we were to mix up the orders If that had happened we would think the slope of the line is positive 12 since we would 2 either have calculated 2 4 or 4 Since we already have a graph of the line and can see it has a negative slope this is clearly incorrect If we don t have a graph to check our calculation we could think about how the point on the left 1 5 is higher than the point on the right 5 3 meaning the slope of the line must be negative 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 Name G8M3 LESSON 10 Date GRADE 8 MISSION 3 LESSON 10 Exit Ticket Without graphing find the slope of the line that goes through 1 0 5 and 8 2 2 2 1 and 6 1 3 3 2 and 1 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 259

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ZEARN MATH STUDENT EDITION G8M3 LESSON 11 Lesson 11 Equations of All Kinds of Lines Let s write equations for vertical and horizontal lines Warm Up Which line doesn t belong 1 A B y y 25 25 20 20 15 15 10 10 5 5 5 C 10 15 20 25 30 x D y 25 20 20 15 15 10 10 5 5 10 15 20 25 30 x 10 15 20 25 30 5 10 15 20 25 30 x y 25 5 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 x 261

G8M3 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Use the graph below to answer the questions y 5 4 3 2 1 5 4 3 2 1 1 1 2 3 4 5 x 2 3 4 5 1 Plot at least 10 points whose y coordinate is 4 What do you notice about them 2 Which equation makes the most sense to represent all of the points with y coordinate 4 Explain how you know x 4 3 262 y 4x y 4 x y 4 Plot at least 10 points whose x coordinate is 3 What do you notice about them 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 4 G8M3 LESSON 11 Which equation makes the most sense to represent all of the points with x coordinate 3 Explain how you know x 3 y 3x 5 Graph the equation x 2 6 Graph the equation y 5 y 3 x y 3 ACTIVITY 2 3 1 There are many possible rectangles whose perimeter is 50 units Solve the following problems Complete the table with lengths l and widths w of at least 10 such rectangles l w 2 The graph shows one rectangle whose perimeter is 50 units and has its lower left vertex at the origin and two sides on the axes On the same graph draw more rectangles with perimeter 50 units using the values from your table Make sure that each rectangle has a lower left vertex at the origin and two sides on the axes y length units 25 20 15 10 5 5 10 15 20 width units 25 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 263

G8M3 LESSON 11 ZEARN MATH STUDENT EDITION 1 Each rectangle has a vertex that lies in the first quadrant These vertices lie on a line Draw in this line and write an equation for it 2 What is the slope of this line How does the slope describe how the width changes as the length changes or vice versa Lesson Summary Horizontal lines in the coordinate plane represent situations where the y value doesn t change at all while the x value changes For example the horizontal line that goes through the point 0 13 can be described in words as for all points on the line the y value is always 13 An equation that says the same thing is y 13 Vertical lines represent situations where the x value doesn t change at all while the y value changes The equation x 4 describes a vertical line through the point 4 0 264 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 G8M3 LESSON 11 Name Date GRADE 8 MISSION 3 LESSON 11 Exit Ticket Here are 5 lines on a coordinate grid a e y 10 b 5 c 10 5 5 10 x d 5 Write equations for lines a b c d and e 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 G8M3 LESSON 12 Lesson 12 Solutions to Linear Equations Let s think about what it means to be a solution to a linear equation with two variables in it Warm Up 1 Which figure has the largest shaded region A B C Concept Exploration ACTIVITY 1 2 1 At the corner produce market apples cost 1 each and oranges cost 2 each Find the cost of a 6 apples and 3 oranges b 4 apples and 4 oranges c 5 apples and 4 oranges d 8 apples and 2 oranges 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

G8M3 LESSON 12 268 ZEARN MATH STUDENT EDITION 2 Noah has 10 to spend at the produce market Can he buy 7 apples and 2 oranges Explain or show your reasoning 3 What combinations of apples and oranges can Noah buy if he spends all of his 10 4 Use two variables to write an equation that represents 10 combinations of apples and oranges Be sure to say what each variable means 5 What are 3 combinations of apples and oranges that make your equation true What are three combinations of apples and oranges that make it 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 G8M3 LESSON 12 ACTIVITY 2 3 You have two numbers If you double the first number and add it to the second number the sum is 10 1 Let x represent the first number and let y represent the second number Write an equation showing the relationship between x y and 10 2 Draw and label a set of x and y axes Plot at least five points on this coordinate plane that make the statement and your equation true What do you notice about the points you have plotted 3 List ten points that do not make the statement true Using a different color plot each point in the same coordinate plane What do you notice about these points compared to your first set of points 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

G8M3 LESSON 12 ZEARN MATH STUDENT EDITION Lesson Summary Think of all the rectangles whose perimeters are 8 units If x represents the width and y represents the length then 2x 2y 8 expresses the relationship between the width and length for all such rectangles For example the width and length could be 1 and 3 since 2 1 2 3 8 or the width and length could be 2 75 and 1 25 since 2 2 75 2 1 25 8 We could find many other possible pairs of width and length x y that make the equation true that is pairs x y that when substituted into the equation make the left side and the right side equal A solution to an equation with two variables is any pair of values x y that make the equation true We can think of the pairs of numbers that are solutions of an equation as points on the coordinate plane Here is a line created by all the points x y that are solutions to 2x 2y 8 Every point on the line represents a rectangle whose perimeter is 8 units All points not on the line are not solutions to 2x 2y 8 y 5 4 3 1 3 2 2 75 1 25 1 1 2 3 4 5 x TERMINOLOGY Solution to an equation with two variables 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 Name G8M3 LESSON 12 Date GRADE 8 MISSION 3 LESSON 12 Exit Ticket Which of the following coordinate pairs make the equation x 9y 12 true 1 12 0 2 0 12 3 3 1 4 0 43 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

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ZEARN MATH STUDENT EDITION G8M3 LESSON 13 Lesson 13 More Solutions to Linear Equations Let s find solutions to more linear equations Warm Up 1 For each equation choose a value for x and then solve to find the corresponding y value that makes that equation true 1 6x 7y 2 5x 3y 9 3 y 5 13 x 7 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

G8M3 LESSON 13 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Here are graphs representing three linear relationships These relationships could also be represented with equations For each statement below decide if it is true or false Explain your reasoning 2 y m 1 4 0 is a solution of the equation for line m 6 n 4 D 2 4 2 H E 2 4 2 The coordinates of the point G make both the equation for line m and the equation for line n true G J K 2 A 4 6 x 3 x 0 is a solution of the equation for line n 4 2 0 makes both the equation for line m and the equation for line n true 5 There is no solution for the equation for line that has y 0 6 The coordinates of point H are a solution to the equation for line 7 There are exactly two solutions of the equation for line 8 There is a point whose coordinates make the equations of all three lines true After you finish discussing the eight statements find another group and check your answers against theirs Discuss any disagreements 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 G8M3 LESSON 13 ACTIVITY 2 3 One partner has 6 cards labeled A through F and one partner has 6 cards labeled a through f In each pair of cards for example Cards A and a there is an equation on one card and a coordinate pair x y that makes the equation true on the other card 1 The partner with the equation asks the partner with a solution for either the x value or the y value and explains why they chose the one they did 2 The partner with the equation uses this value to find the other value explaining each step as they go 3 The partner with the coordinate pair then tells the partner with the equation if they are right or wrong If they are wrong both partners should look through the steps to find and correct any errors If they are right both partners move on to the next set of cards 4 Keep playing until you have finished Cards A through 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 275

G8M3 LESSON 13 ZEARN MATH STUDENT EDITION Lesson Summary Let s think about the linear equation 2x 4y 12 If we know 0 3 is a solution to the equation then we also know 0 3 is a point on the graph of the equation Since this point is on the y axis we also know that it is the vertical intercept of the graph But what about the coordinate of the horizontal intercept when y 0 Well we can use the equation to figure it out 2x 4y 12 2x 4 0 12 2x 12 x 6 Since x 6 when y 0 we know the point 6 0 is on the graph of the line No matter the form a linear equation comes in we can always find solutions to the equation by starting with one value and then solving for the other value 276 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 G8M3 LESSON 13 Date GRADE 8 MISSION 3 LESSON 13 Exit Ticket A graph of a linear equation passes through 2 0 and 0 6 1 Use the two points to sketch the graph of the equation 2 Is 3x y 6 an equation for this graph 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 277

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ZEARN MATH STUDENT EDITION G8M3 LESSON 14 Lesson 14 Using Linear Relations to Solve Problems Let s write equations for real world situations and think about their solutions Warm Up 1 For each relationship described write an equation to represent the relationship 1 Grapes cost 2 39 per pound Bananas cost 0 59 per pound You spend 15 on g pounds of grapes and b pounds of bananas 2 A savings account has 50 in it at the start of the year and 20 is deposited each week After x weeks there are y dollars in the account 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

G8M3 LESSON 14 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 280 Each line represents one person s weekly savings account balance from the start of the year 1 Choose one line and write a description of what happens to that person s account over the first 17 weeks of the year Do not tell your group which line you chose 2 Share your story with your group and see if anyone can guess your line 3 Write an equation for each line on the graph What do the slope m and vertical intercept b in each equation mean in the situation 4 For which equation is 1 70 a solution Interpret this solution in terms of your story 5 Predict the balance in each account after 20 weeks 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 G8M3 LESSON 14 ACTIVITY 2 3 The Fabulous Fish Market orders tilapia which costs 3 per pound and salmon which costs 5 per pound The market spends 210 on this order each day 1 What are five different combinations of salmon and tilapia that the market can order 2 Define variables and write an equation representing the relationship between the amount of each fish bought and how much the market spends 3 Sketch a graph of the relationship Label your axes 4 On your graph plot and label the combinations A F A B C D E F Pounds of tilapia 5 19 27 25 65 55 Pounds of salmon 36 30 6 25 27 6 4 a Which of these combinations can the market order Explain or show your reasoning 5 List two ways you can tell if a pair of numbers is a solution to an 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 281

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ZEARN MATH STUDENT EDITION G8M3 LESSON 15 Lesson 15 Linear Inequalities in Two Variables Let s find out how to use graphs to represent solutions to inequalities in two variables Warm Up 1 Here is an expression 2x 3y Decide if the values in each ordered pair x y make the value of the expression less than greater than or equal to 12 0 5 6 0 1 1 5 10 Concept Exploration ACTIVITY 1 2 Work with your group to study each inequality assigned to your group and answer the questions below Find some coordinate pairs that represent solutions to the inequality and some coordinate pairs that do not represent solutions Plot both sets of points Either use two different colors or two different symbols like X and O Plot enough points until you start to see the region that contains solutions and the region that contains non solutions Look for a pattern describing the region where solutions are plotted 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

G8M3 LESSON 15 284 ZEARN MATH STUDENT EDITION x y 2y 4 3x 0 x y 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 G8M3 LESSON 15 ACTIVITY 2 3 Here is a graph that represents solutions to the equation x y 5 Follow the directions below Sketch 4 quick graphs representing the solutions to each of these inequalities x y5 x y 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 285

G8M3 LESSON 15 43 286 ZEARN MATH STUDENT EDITION For each graph write an inequality whose solutions are represented by the shaded part of the graph 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

ZEARN MATH STUDENT EDITION G8M3 LESSON 15 Lesson Summary The equation x y 7 is an equation in two variables Its solution is any pair of x and y whose sum is 7 The pairs x 0 y 7 and x 5 y 2 are two examples We can represent all the solutions to x y 7 by graphing the equation on a coordinate plane The graph is a line All the points on the line are solutions to x y 7 The inequality x y 7 is an inequality in two variables Its solution is any pair of x and y whose sum is 7 or less than 7 This means it includes all the pairs that are solutions to the equation x y 7 but also many other pairs of x and y that add up to a value less than 7 The pairs x 4 y 7 and x 6 y 0 are two examples On a coordinate plane the solution to x y 7 includes the line that represents x y 7 If we plot a few other x y pairs that make the inequality true such as 4 7 and 6 0 we see that these points fall on one side of the line In contrast x y pairs that make the inequality false fall on the other side of the line We can shade that region on one side of the line to indicate that all points in it are solutions What about the inequality x y 7 The solution is any pair of x and y whose sum is less than 7 This means pairs like x 0 y 7 and x 5 y 2 are not solutions On a coordinate plane the solution does not include points on the line that represent x y 7 because those points are x and y pairs whose sum is 7 To exclude points on that boundary line we can use a dashed line All points below that line are x y pairs that make x y 7 true The region on that side of the line can be shaded to show that it contains the 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 287

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ZEARN MATH STUDENT EDITION G8M3 LESSON 15 Name Date GRADE 8 MISSION 3 LESSON 15 Exit Ticket The line in each graph represents y 2x Which graph represents 2x y 1 A B 8 6 4 y y 8 8 6 6 4 4 2 2 2 2 4 6 8 x 6 4 2 2 2 4 4 6 6 8 8 C 2 4 6 8 x 2 4 6 8 x D 8 2 8 6 4 y y 8 8 6 6 4 4 2 2 2 2 4 6 8 x 8 6 4 2 2 2 4 4 6 6 8 8 Explain your reasons for choosing that 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 289

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ZEARN MATH STUDENT EDITION G8V1 Terminology Alternate interior angles transversal Alternate interior angles are created when two parallel lines are crossed by another line called a transversal Alternate interior angles are inside the parallel lines and on opposite sides of the transversal a c This diagram shows two pairs of alternate interior angles Angles a and d are one pair and angles b and c are another pair b d Center of a dilation The center of a dilation is a fixed point on a plane It is the starting point from which we measure distances in a dilation In this diagram point P is the center of the dilation P A B C Clockwise Clockwise means to turn in the same direction as the hands of a clock It is a turn to the right B This diagram shows Figure A turned clockwise to make Figure B A Congruent One figure is congruent to another if it can be moved with translations rotations and reflections to fit exactly over the other C A D B In the figure Triangle A is congruent to Triangles B C and D A translation takes Triangle A to Triangle B a rotation takes Triangle B to Triangle C and a reflection takes Triangle C to Triangle 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 291

G8V1 ZEARN MATH STUDENT EDITION Corresponding When part of an original figure matches up with part of a copy we call them corresponding parts These could be points segments angles or distances A C D F For example point B in the first triangle corresponds to point E in the second triangle B Segment AC corresponds to segment DF E Counterclockwise Counterclockwise means to turn opposite of the way the hands of a clock turn It is a turn to the left This diagram shows Figure A turned counterclockwise to make Figure B B A Dilation A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point The fixed point is the center of the dilation All of the original distances are multiplied by the same scale factor For example triangle DEF is a dilation of triangle ABC The fixed center point is O and the scale factor is 3 E F D B C A O This means that every point of triangle DEF is 3 times as far from O as every corresponding point of triangle ABC Image F An image is the result of translations rotations and reflections on an object Every part of the original object moves in the same way to match up with a part of the image E C In this diagram triangle ABC has been translated up and to the right to make triangle DEF Triangle DEF is the image of the original triangle ABC D B A 292 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 G8V1 y Linear relationship The rate of change in a linear relationship is the amount y changes when x increases by 1 The rate of change in a linear relationship is also the slope of its graph In this graph y increases by 15 dollars when x increases by 1 hour The rate of change is 15 dollars per hour 140 120 100 80 60 40 20 1 2 3 4 Number of days 5 x 160 140 Amount earned dollars Rate of change 160 Number of pages read A linear relationship between two quantities means they are related like this When one quantity changes by a certain amount the other quantity always changes by a set amount In a linear relationship one quantity has a constant rate of change with respect to the other The relationship is called linear because its graph is a line The graph shows a relationship between number of days and number of pages read When the number of days increases by 2 the number of pages read always increases by 60 The rate of change is constant 30 pages per day so the relationship is linear 120 100 80 60 40 20 1 2 3 4 5 6 Time hours 7 8 9 Reflection A reflection across a line moves every point on a figure to a point directly on the opposite side of the line The new point is the same distance from the line as it was in the original figure This diagram shows a reflection of A over line that makes the mirror image B B A Rigid transformation A rigid transformation is a move that does not change any measurements of a figure Translations rotations and reflections are rigid transformations as is any sequence of these Rotation A rotation moves every point on a figure around a center by a given angle in a specific direction This diagram shows Triangle A rotated around center O by 55 degrees clockwise to get Triangle B B A 55 O 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

G8V1 ZEARN MATH STUDENT EDITION Sequence of transformations A sequence of transformations is a set of translations rotations reflections and dilations on a figure The transformations are performed in a given order P Q B A This diagram shows a sequence of transformations to move Figure A to Figure C First A is translated to the right to make B Next B is reflected across line to make C C R Similar Two figures are similar if one can fit exactly over the other after rigid transformations and dilations In this figure triangle ABC is similar to triangle DEF E F C If ABC is rotated around point B and B then dilated with center point O then it will fit exactly over DEF This means that they are similar O Slope The slope of a line is a number we can calculate using any two points on the line To find the slope divide the vertical distance between the points by the horizontal distance The slope of this line is 2 divided by 3 or 23 D A y 4 3 Vertical distance 2 1 Horizontal distance 1 2 3 4 x Solution to an equation with two variables A solution to an equation with two variables is a pair of values of the variables that make the equation true For example one possible solution to the equation 4x 3y 24 is 6 0 Substituting 6 for x and 0 for y makes this equation true because 4 6 3 0 24 Straight angle A straight angle is an angle that forms a straight line It measures 180 degrees 294 straight angle 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 G8V1 Translation A translation moves every point in a figure a given distance in a given direction B This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow A Transversal A transversal to two parallel lines is a line that cuts across them intersecting each one m This diagram shows a transversal line k intersecting parallel lines m and k Vertical angles Vertical angles are opposite angles that share the same vertex They are formed by a pair of intersecting lines Their angle measures are equal C A E For example angles AEC and DEB are vertical angles If angle AEC measures 120 then angle DEB must also measure 120 Angles AED and BEC are another pair of vertical angles B D y Vertical intercept 10 The vertical intercept is the point where the graph of a line crosses the vertical axis The vertical intercept of this line is 0 6 or just 6 8 6 4 2 10 8 6 4 2 2 2 4 6 8 10 x 4 6 8 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 295

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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 Vol 2 Mission 4 Linear Equations and Linear Systems Mission 5 Functions and Volume Mission 6 Associations in Data Student Edition Mission 3 Linear Relationships Vol 3 Mission 7 Exponents and Scientific Notation Mission 8 Pythagorean Theorem and Irrational Numbers Mission 9 Putting It All Together G8 Vol 1 Zearnmath_SE_Grade8_Vol1 indd 1 Grade 8 Volume 1 MISSIONS 1 2 3 4 5 6 7 8 9 12 15 22 1 35 PM