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STUDENT EDITION Grade 7 VOLUME 1 Mission 1 Scale Drawings Mission 2 Introducing Proportional Relationships Mission 3 Measuring Circles NAME

2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum used under the CC BY 4 0 license Download the original for free at openupresources org Zearn is a registered trademark Printed in the U S A ISBN 978 8 88868 887 8

Table of Contents Mission 1 Lesson 1 What are Scaled Copies 3 Lesson 2 Corresponding Parts and Scale Factors 9 Lesson 3 Making Scaled Copies 15 Lesson 4 Scaled Relationships 21 Lesson 5 The Size of the Scale Factor 27 Lesson 6 Scaling and Area 31 Lesson 7 Scale Drawings 37 Lesson 8 Scale Drawings and Maps 43 Lesson 9 Creating Scale Drawings 49 Lesson 10 Changing Scales in Scale Drawings 55 Lesson 11 Scales Without Units 61 Lesson 12 Units in Scale Drawings 67 Lesson 13 Draw It to Scale 73 Mission 2 Lesson 1 One of These Things Is Not Like the Others 77 Lesson 2 Introducing Proportional Relationships with Tables 83 Lesson 3 More about Constant of Proportionality 89 Lesson 4 Proportional Relationships and Equations 95 Lesson 5 Two Equations for Each Relationship 101 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

Lesson 6 Using Equations to Solve Problems 107 Lesson 7 Comparing Relationships with Tables 113 Lesson 8 Comparing Relationships with Equations 119 Lesson 9 Solving Problems about Proportional Relationships 125 Lesson 10 Introducing Graphs of Proportional Relationships 129 Mission 3 Lesson 1 How Well Can You Measure 137 Lesson 2 Exploring Circles 143 Lesson 3 Exploring Circumference 149 Lesson 4 Applying Circumference 155 Lesson 5 Circumference and Wheels 161 Lesson 6 Estimating Areas 167 Lesson 7 Exploring the Area of a Circle 173 Lesson 8 Relating Area to Circumference 179 Lesson 9 Applying Area of Circles 183 Lesson 10 Distinguishing Circumference and Area 189 Lesson 11 Stained Glass Windows 195 Terminology 197 iv 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Grade 7 Mission 1 Scale Drawings

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ZEARN MATH STUDENT EDITION G7M1 LESSON 1 Lesson 1 What are Scaled Copies Let s explore scaled copies Warm Up 1 1 Here is a picture of a bird Look at pictures A E How is each one the same as or different from the original picture of the bird A B C D E 2 Some of the pictures A E are scaled copies of the original picture Which ones do you think are scaled copies Explain your reasoning 3 What do you think scaled copy means 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 On the top left is the original drawing of the letter F There are also several other drawings Identify all the drawings that are scaled copies of the original letter F drawing Explain how you know Original Drawing 1 Drawing 2 Drawing 3 Drawing 4 Drawing 5 Drawing 6 Drawing 7 2 Examine all the scaled copies more closely specifically the lengths of each part of the letter F How do they compare to the original What do you notice 3 4 On the grid draw a different scaled copy of the original letter 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

ZEARN MATH STUDENT EDITION G7M1 LESSON 1 ACTIVITY 2 43 1 Your teacher will give you a set of cards that have polygons drawn on a grid Mix up the cards and place them all face up Take turns with your partner to match a pair of polygons that are scaled copies of one another a For each match you find explain to your partner how you know it s a match b For each match your partner finds listen carefully to their explanation and if you disagree explain your thinking 2 When you agree on all of the matches check your answers with the answer key If there are any errors discuss why and revise your matches 3 Select one pair of polygons to examine further Draw both polygons on the grid Explain how you know that one polygon is a scaled copy of the other 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary What is a scaled copy of a figure Let s look at some examples Original The second and third drawings are both scaled copies of the original Y However here the second and third drawings are not scaled copies of the original W Original The second drawing is spread out wider and shorter The third drawing is squished in narrower but the same height We will learn more about what it means for one figure to be a scaled copy of another in upcoming lessons TERMINOLOGY Scaled copy 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M1 LESSON 1 Name Date GRADE 7 MISSION 1 LESSON 1 Exit Ticket 1 Are any of the figures B C or D scaled copies of figure A Explain how you know A C Original B 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 7

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ZEARN MATH STUDENT EDITION G7M1 LESSON 2 Lesson 2 Corresponding Parts and Scale Factors Let s describe features of scaled copies Warm Up 1 1 1 4 Find each product mentally 32 2 7 2 1 9 3 1 4 5 6 Concept Exploration ACTIVITY 1 2 Here is a figure and two copies each with some points labeled K U A L M N E C D P Original 1 V B O F W Y X Copy 1 Z Copy 2 Complete this table to show corresponding parts in the three figures Original Copy 1 Copy 2 Point P Segment LM Segment EF Point W Angle KLM Angle 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 9

G7M1 LESSON 2 ZEARN MATH STUDENT EDITION 2 Is either copy a scaled copy of the original figure Explain your reasoning 3 Use tracing paper to compare angle KLM with its corresponding angles in Copy 1 and Copy 2 What do you notice 4 Use tracing paper to compare angle NOP with its corresponding angles in Copy 1 and Copy 2 What do you notice ACTIVITY 2 3 Here is Triangle O followed by a number of other triangles O A 5 4 B 6 08 10 8 H 2 G 8 7 2 83 2 2 F 3 2 5 39 5 3 E 5 2 2 5 4 6 32 3 3 D C 10 3 8 3 2 6 6 Your teacher will assign you two of the triangles to look at 1 10 For each of your assigned triangles is it a scaled copy of Triangle O Be prepared to explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M1 LESSON 2 2 As a group identify all the scaled copies of Triangle O in the collection Discuss your thinking If you disagree work to reach an agreement 3 List all the triangles that are scaled copies in the table Record the side lengths that correspond to the side lengths of Triangle O listed in each column Triangle O 4 3 4 5 Explain or show how each copy has been scaled from the original Triangle 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 11

G7M1 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary A figure and its scaled copy have corresponding parts or parts that are in the same position in relation to the rest of each figure These parts could be points segments or angles For example Polygon 2 is a scaled copy of Polygon 1 G 5 6 A 2 8 3 B 1 3 D F 2 E 2 3 2 Polygon 1 C 6 4 L J 2 6 K 4 H 6 4 I Polygon 2 Each point in Polygon 1 has a corresponding point in Polygon 2 For example point B corresponds to point H and point C corresponds to point I Each segment in Polygon 1 has a corresponding segment in Polygon 2 For example segment AF corresponds to segment GL Each angle in Polygon 1 also has a corresponding angle in Polygon 2 For example angle DEF corresponds to angle JKL The scale factor between Polygon 1 and Polygon 2 is 2 because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1 The angle measures in Polygon 2 are the same as the corresponding angle measures in Polygon 1 for example the measure of angle JKL is the same as the measure of angle DEF TERMINOLOGY Corresponding Scale factor 12 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M1 LESSON 2 Name Date GRADE 7 MISSION 1 LESSON 2 Exit Ticket Polygon PQRS is a scaled copy of polygon ABCD A D B P S Q C R 1 Name the angle in the scaled copy that corresponds to angle ABC 2 Name the segment in the scaled copy that corresponds to segment AD 3 What is the scale factor from polygon ABCD to polygon PQRS 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M1 LESSON 3 Lesson 3 Making Scaled Copies Let s draw scaled copies Warm Up 1 1 For each problem select the answer from the two choices The value of 25 8 5 is a More than 205 b Less than 205 2 The value of 9 93 0 984 is a More than 10 b Less than 10 3 The value of 0 24 0 67 is a More than 0 2 b Less than 0 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 15

G7M1 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy 36 18 30 12 15 21 Diego and Jada each use a different operation to find the new side lengths Here are their finished drawings 26 12 8 20 10 5 6 4 2 5 11 Diego s Drawing 16 7 Jada s Drawing 1 What operation do you think Diego used to calculate the lengths for his drawing 2 What operation do you think Jada used to calculate the lengths for her drawing 3 Did each method produce a scaled copy of the polygon 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 G7M1 LESSON 3 ACTIVITY 2 3 Andre wants to make a scaled copy of Jada s drawing so the side that corresponds to 4 units in Jada s polygon is 8 units in his scaled copy 12 10 5 6 4 7 Jada s Drawing 1 Andre says I wonder if I should add 4 units to the lengths of all of the segments What would you say in response to Andre Explain or show your reasoning 2 Create the scaled copy that Andre wants If you get stuck consider using the edge of an index card or paper to measure the lengths needed to draw the copy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor For example to make a scaled copy of triangle ABC where the base is 8 units we would use a scale factor of 4 This means multiplying all the side lengths by 4 so in triangle DEF each side is 4 times as long as the corresponding side in triangle ABC F 2 24 A 18 2 8 96 C 1 B D 8 4 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 G7M1 LESSON 3 Name Date GRADE 7 MISSION 1 LESSON 3 Exit Ticket 1 Create a scaled copy of ABCD using a scale factor of 4 B A D C 2 Triangle Z is a scaled copy of Triangle M M 7 4 10 Select all the sets of values that could be the side lengths of Triangle Z a 8 11 and 14 b 10 17 5 and 25 c 6 9 and 11 d 6 10 5 and 15 e 8 14 and 20 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M1 LESSON 4 Lesson 4 Scaled Relationships Let s find relationships between scaled copies Warm Up 1 Each of these polygons is a scaled copy of the others G c D K B H F L J I A E 1 Name two pairs of corresponding angles What can you say about the sizes of these angles 2 Check your prediction by measuring at least one pair of corresponding angles using a protractor Record your measurements to the nearest 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 21

G7M1 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Each of these polygons is a scaled copy of the others You already checked their corresponding angles G c D K B H F L J I A E 1 The side lengths of the polygons are hard to tell from the grid but there are other corresponding distances that are easier to compare Identify the distances in the other two polygons that correspond to DB and AC and record them in the table Quadrilateral Distance that corresponds to DB Distance that corresponds to AC ABCD DB 4 AC 6 EFGH IJKL 2 Look at the values in the table What do you notice Pause here so your teacher can review your work A 3 6 C The larger figure is a scaled copy of the smaller figure a If AE 4 how long is the corresponding distance in the second figure Explain or show your reasoning B D E H L 15 J I K b If IK 5 how long is the corresponding distance in the first figure Explain or show your reasoning 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 G7M1 LESSON 4 ACTIVITY 2 3 1 Here are two more pairs of quadrilaterals Mai says that Polygon ZSCH is a scaled copy of Polygon XJYN but Noah disagrees Do you agree with either of them Explain or show your reasoning N H X Y Z D C 120 4 S 60 J 2 A Record the corresponding distances in the table What do you notice Quadrilateral Horizontal distance Vertical distance XJYN XY JN ZSCH ZC SH 3 Measure at least three pairs of corresponding angles in XJYN and ZSCH using a protractor Record your measurements to the nearest 5 What do you notice 4 Do these results change your answer to the first question Explain D Here are two more quadrilaterals 2 120 120 4 5 H 60 6 1 G 120 120 4 60 A C 3 3 60 B E 60 4 F Kiran says that Polygon EFGH is a scaled copy of ABCD but Lin disagrees Do you agree with either of them 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 23

G7M1 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary When a figure is a scaled copy of another figure we know that 1 A All distances in the copy can be found by multiplying the corresponding distances in the original figure by the same scale factor whether or not the endpoints are connected by a segment F B C D E T For example Polygon STUVWX is a scaled copy of Polygon ABCDEF The scale factor is 3 The distance from T to X is 6 which is three times the distance from B to F 2 X V U W All angles in the copy have the same measure as the corresponding angles in the original figure as in these triangles These observations can help explain why one figure is not a scaled copy of another For example even though their corresponding angles have the same measure the second rectangle is not a scaled copy of the first rectangle because different pairs of corresponding lengths have different scale factors 2 12 1 but 3 23 2 24 S 60 60 42 42 78 78 Original 2 3 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

ZEARN MATH STUDENT EDITION G7M1 LESSON 4 Name Date GRADE 7 MISSION 1 LESSON 4 Exit Ticket Here are two polygons on a grid P A B C E D Q R T S Is PQRST a scaled copy of ABCDE 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 25

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ZEARN MATH STUDENT EDITION G7M1 LESSON 5 Lesson 5 The Size of the Scale Factor Let s look at the effects of different scale factors Warm Up 1 1 Solve each equation mentally 16x 176 2 16x 8 3 16x 1 4 1 5 x 1 5 2 5 x 1 Concept Exploration ACTIVITY 1 2 You will receive a set of cards On each card Figure A is the original and Figure B is a scaled copy 1 Sort the cards based on their scale factors Be prepared to explain your reasoning 2 Examine cards 10 and 13 more closely What do you notice about the shapes and sizes of the figures What do you notice about the scale factors 3 Examine cards 8 and 12 more closely What do you notice about the figures What do you notice about the scale factors 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 5 ZEARN MATH STUDENT EDITION Lesson Summary The size of the scale factor affects the size of the copy When a figure is scaled by a scale factor greater than 1 the copy is larger than the original When the scale factor is less than 1 the copy is smaller When the scale factor is exactly 1 the copy is the same size as the original Triangle DEF is a larger scaled copy of triangle ABC because the scale factor from ABC to DEF is 32 Triangle ABC is a smaller scaled copy of triangle DEF because the scale factor from DEF to ABC is 23 C 5 A F 3 2 7 5 4 5 3 4 B D 6 E 2 3 This means that triangles ABC and DEF are scaled copies of each other It also shows that scaling can be reversed using reciprocal scale factors such as 23 and 32 In other words if we scale Figure A using a scale factor of 4 to create Figure B we can scale Figure B using the reciprocal scale factor 14 to create Figure A 28 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 5 Date GRADE 7 MISSION 1 LESSON 5 Exit Ticket A rectangle that is 2 inches by 3 inches has been scaled by a factor of 7 1 What are the side lengths of the scaled copy 2 Suppose you want to scale the copy back to its original size What scale factor should 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 29

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ZEARN MATH STUDENT EDITION G7M1 LESSON 6 Lesson 6 Scaling and Area Let s build scaled shapes and investigate their areas Warm Up 1 Work with your group to build the scaled copies described in each question A 1 B C How many blue rhombus blocks does it take to build a scaled copy of Figure A a Where each side is twice as long b Where each side is 3 times as long c 2 Where each side is 4 times as long How many green triangle blocks does it take to build a scaled copy of Figure B a Where each side is twice as long b Where each side is 3 times as long c 3 Using a scale factor of 4 How many red trapezoid blocks does it take to build a scaled copy of Figure C a Using a scale factor of 2 b Using a scale factor of 3 c Using a scale factor of 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 31

G7M1 LESSON 6 4 ZEARN MATH STUDENT EDITION Make a prediction How many blocks would it take to build scaled copies of these shapes using a scale factor of 5 Using a scale factor of 6 Be prepared to explain your reasoning Concept Exploration ACTIVITY 1 2 Follow the directions and answer the questions about your group s assigned figure D 32 E F 1 Build a scaled copy of your assigned shape using a scale factor of 2 Use the same shape of blocks as in the original figure How many blocks did it take 2 Your classmate thinks that the scaled copies in the previous problem will each take 4 blocks to build Do you agree or disagree Explain your reasoning 3 Start building a scaled copy of your assigned figure using a scale factor of 3 Stop when you can tell for sure how many blocks it would take Record your answer 4 How many blocks would it take to build scaled copies of your figure using scale factors 4 5 and 6 Explain or show your reasoning 5 How is the pattern in this activity the same as the pattern you saw in the previous activity How is it 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 G7M1 LESSON 6 ACTIVITY 2 3 Answer the questions about your shape 1 Your teacher will give you a figure with measurements in centimeters What is the area of your figure How do you know 2 Work with your partner to draw scaled copies of your figure using each scale factor in the table Complete the table with the measurements of your scaled copies Scale factor Base cm Height cm Area cm2 1 2 3 1 2 1 3 3 Compare your results with a group that worked with a different figure What is the same about your answers What is different 4 If you drew scaled copies of your figure with the following scale factors what would their areas be Discuss your thinking If you disagree work to reach an agreement Be prepared to explain your reasoning Scale factor Area cm2 5 3 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 33

G7M1 LESSON 6 ZEARN MATH STUDENT EDITION Lesson Summary Scaling affects lengths and areas differently When we make a scaled copy all original lengths are multiplied by the scale factor If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3 the side lengths of the copy will be 6 units and 12 units because 2 3 6 and 4 3 12 6 2 12 4 The area of the copy however changes by a factor of scale factor 2 If each side length of the copy is 3 times longer than the original side length then the area of the copy will be 9 times the area of the original because 3 3 or 32 equals 9 6 2 4 12 In this example the area of the original rectangle is 8 units2 and the area of the scaled copy is 72 units2 because 9 8 72 We can see that the large rectangle is covered by 9 copies of the small rectangle without gaps or overlaps We can also verify this by multiplying the side lengths of the large rectangle 6 12 72 Lengths are one dimensional so in a scaled copy they change by the scale factor Area is twodimensional so it changes by the square of the scale factor We can see this is true for a rectangle with length l and width w If we scale the rectangle by a scale factor of s we get a rectangle with length s l and width s w The area of the scaled rectangle is A s l s w so A s2 l w The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two dimensional figures too not just for rectangles 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 G7M1 LESSON 6 Name Date GRADE 7 MISSION 1 LESSON 6 Exit Ticket 1 Lin has a drawing with an area of 20 in2 If she increases all the sides by a scale factor of 4 what will the new area be 20 in2 2 Noah enlarged a photograph by a scale factor of 6 The area of the enlarged photo is how many times as large as the area of the original 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 7 Lesson 7 Scale Drawings Let s explore scale drawings Warm Up 1 Here are some drawings of a school bus a quarter and the subway lines around Boston Massachusetts The first three drawings are scale drawings of these objects The next three drawings are not scale drawings of these objects Discuss with your partner what a scale drawing is Concept Exploration ACTIVITY 1 2 You will receive a scale drawing of a basketball court The drawing does not have any measurements labeled but it says that 1 cm represents 2 meters 1 Measure the distances on the scale drawing that are labeled a d to the nearest tenth of a centimeter Record your results in the first row of the table 2 The statement 1 cm represents 2 m is the scale of the drawing It can also be expressed as 1 cm to 2 m or 1 cm for every 2 m What do you think the scale tells us 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 7 3 ZEARN MATH STUDENT EDITION How long would each measurement from the first question be on an actual basketball court Explain or show your reasoning a length of court b width of court c hoop to hoop d 3 point line to sideline Scale drawing Actual court 4 On an actual basketball court the bench area is typically 9 meters long a Without measuring determine how long the bench area should be on the scale drawing b Check your answer by measuring the bench area on the scale drawing Did your prediction match your measurement ACTIVITY 2 3 1 38 Here is a scale drawing of some of the world s tallest structures About how tall is the actual Willis Tower About how tall is the actual Great Pyramid Be prepared to explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M1 LESSON 7 2 About how much taller is the Burj Khalifa than the Eiffel Tower Explain or show your reasoning 3 Measure the line segment that shows the scale to the nearest tenth of a centimeter Express the scale of the drawing using numbers and words 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 39

G7M1 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary Scale drawings are two dimensional representations of actual objects or places Floor plans and maps are some examples of scale drawings On a scale drawing Every part corresponds to something in the actual object Lengths on the drawing are enlarged or reduced by the same scale factor A scale tells us how actual measurements are represented on the drawing For example if a map has a scale of 1 inch to 5 miles then a 12 inch line segment on that map would represent an actual distance of 2 5 miles Sometimes the scale is shown as a segment on the drawing itself For example here is a scale drawing of a stop sign with a line segment that represents 25 cm of actual length The width of the octagon in the drawing is about three times the length of this segment so the actual width of the sign is about 3 25 or 75 cm Because a scale drawing is two dimensional some aspects of the threedimensional object are not represented For example this scale drawing does not show the thickness of the stop sign A scale drawing may not show every detail of the actual object however the features that are shown correspond to the actual object and follow the specified scale TERMINOLOGY Scale Scale Drawing 40 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 7 Date GRADE 7 MISSION 1 LESSON 7 Exit Ticket 1 A scale drawing of a school bus has a scale of 12 inch to 5 feet If the length of the school bus is 4 12 inches on the scale drawing what is the actual length of the bus Explain or show your reasoning 2 A scale drawing of a lake has a scale of 1 cm to 80 m If the actual width of the lake is 1 000 m what is the width of the lake on the scale drawing 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 41

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ZEARN MATH STUDENT EDITION G7M1 LESSON 8 Lesson 8 Scale Drawings and Maps Let s use scale drawings to solve problems Warm Up 1 Use the information below to determine which vehicle is traveling faster Two cities are 243 miles apart It takes a train 4 hours to travel between the two cities at a constant speed A car travels between the two cities at a constant speed of 65 miles per hour Which is traveling faster the car or the train Be prepared to explain your reasoning Concept Exploration ACTIVITY 1 2 1 Use the map of Interstate 90 outside of Chicago to answer the following questions A driver is traveling at a constant speed on Interstate 90 outside Chicago If she traveled from Point A to Point B in 8 minutes did she obey the speed limit of 55 miles per hour 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 43

G7M1 LESSON 8 2 ZEARN MATH STUDENT EDITION A traffic helicopter flew directly from Point A to Point B in 8 minutes Did the helicopter travel faster or slower than the driver Explain your reasoning ACTIVITY 2 3 A cyclist rides at a constant speed of 15 miles per hour At this speed about how long would it take the cyclist to ride from Garden City to Dodge City Kansas Map of Kansas by United States Census Bureau via American Fact Finder Public Domain 44 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M1 LESSON 8 Lesson Summary Maps with scales are useful for making calculations involving speed time and distance Here is a map of part of Alabama Suppose it takes a car 1 hour and 30 minutes to travel at constant speed from Birmingham to Montgomery How fast is the car traveling To make an estimate we need to know about how far it is from Birmingham to Montgomery The scale of the map represents 20 miles so we can estimate the distance between these cities is about 90 miles 2 13 Time Hours Distance Miles 1 5 90 3 180 1 60 2 13 Since 90 miles in 1 5 hours is the same speed as 180 miles in 3 hours the car is traveling about 60 miles per hour Suppose a car is traveling at a constant speed of 60 miles per hour from Montgomery to Centreville How long will it take the car to make the trip Using the scale we can estimate that it is about 70 miles Since 60 miles per hour is the same as 1 mile per minute it will take the car about 70 minutes or 1 hour and 10 minutes to make this trip 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 45

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ZEARN MATH STUDENT EDITION G7M1 LESSON 8 Name Date GRADE 7 MISSION 1 LESSON 8 Exit Ticket Here is a map of the Missouri Botanical Garden Clare walked all the way around the garden Map of Missouri Botanical Garden by United States Census Bureau via American Fact Finder Public Domain 1 What is the actual distance around the garden Show your reasoning 2 It took Clare 30 minutes to walk around the garden at a constant speed At what speed was she walking 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 47

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ZEARN MATH STUDENT EDITION G7M1 LESSON 9 Lesson 9 Creating Scale Drawings Let s create our own scale drawings Warm Up 1 Without calculating decide which quotient is larger 1 11 23 or 7 13 2 0 63 2 or 0 55 3 3 15 1 3 or 15 1 4 Concept Exploration ACTIVITY 1 2 Here is a rough sketch of Noah s bedroom not a scale drawing Noah wants to create a floor plan that is a scale drawing Wall E Wall A Wall B Wall D Wall C 1 The actual length of Wall C is 4 m To represent Wall C Noah draws a segment 16 cm long What scale is he using Explain your reasoning 2 Find another way to express the scale 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 9 ZEARN MATH STUDENT EDITION 3 Discuss your thinking with your partner How do your scales compare 4 The actual lengths of Wall A and Wall D are 2 5 m and 3 75 m Determine how long these walls will be on Noah s scale floor plan Explain your reasoning ACTIVITY 2 3 50 A rectangle around Utah is about 270 miles wide and about 350 miles tall The upper right corner that is missing is about 110 miles wide and about 70 miles tall Make two scale drawings of Utah 1 Make a scale drawing of Utah where 1 centimeter represents 50 miles 2 Make a scale drawing of Utah where 1 centimeter represents 75 miles 3 How do the two drawings compare How does the choice of scale influence the drawing 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 9 Lesson Summary If we want to create a scale drawing of a room s floor plan that has the scale 1 inch to 4 feet we can divide the actual lengths in the room in feet by 4 to find the corresponding lengths in inches for our drawing Suppose the longest wall is 15 feet long We should draw a line 3 75 inches long to represent this wall because 15 4 3 75 1 inch 4 feet There is more than one way to express this scale These three scales are all equivalent since they represent the same relationship between lengths on a drawing and actual lengths 1 inch to 4 feet 1 2 inch to 2 feet 1 4 inch to 1 foot Any of these scales can be used to find actual lengths and scaled lengths lengths on a drawing For instance we can tell that at this scale an 8 foot long wall should be 2 inches long on the drawing because 14 8 2 The size of a scale drawing is influenced by the choice of scale For example here is another scale drawing of the same room using the scale 1 inch to 8 feet Notice this drawing is smaller than the previous one Since one inch on this drawing represents twice as much actual distance each side length only needs to be half as long as it was in the first scale drawing 1 inch 8 feet 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 51

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ZEARN MATH STUDENT EDITION Name G7M1 LESSON 9 Date GRADE 7 MISSION 1 LESSON 9 Exit Ticket A rectangular swimming pool measures 50 meters in length and 25 meters in width 1 Make a scale drawing of the swimming pool where 1 centimeter represents 5 meters 2 What are the length and width of your scale drawing 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M1 LESSON 10 Lesson 10 Changing Scales in Scale Drawings Let s explore different scale drawings of the same actual thing Warm Up 1 If a student uses a ruler like this to measure the length of their foot which choices would be appropriate measurements Select all that apply Be prepared to explain your reasoning Inches Centimeters 1 a 9 inches b 1 4 5 9 64 inches c 23 47659 centimeters 1 2 2 3 4 5 3 6 7 8 9 10 d 23 5 centimeters e 23 48 centimeters 2 Here is a scale drawing of an average seventh grade student s foot next to a scale drawing of a foot belonging to the person with the largest feet in the world Estimate the length of the larger foot 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 Here is a map showing a plot of land in the shape of a right triangle Use the map to answer the questions Logan Square map by United States Census Bureau via American Fact Finder Public Domain 56 1 Your teacher will assign you a scale to use On centimeter graph paper make a scale drawing of the plot of land Make sure to write your scale on your drawing 2 What is the area of the triangle you drew Explain or show your reasoning 3 How many square meters are represented by 1 square centimeter in your drawing 4 After everyone in your group is finished order the scale drawings from largest to smallest What do you notice about the scales when your drawings are placed in this 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

ZEARN MATH STUDENT EDITION G7M1 LESSON 10 ACTIVITY 2 43 Here is a scale drawing of a playground The scale is 1 centimeter to 30 meters Answer the questions B C A D 1 Make another scale drawing of the same playground at a scale of 1 centimeter to 20 meters 2 How do the two scale drawings compare 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 10 ZEARN MATH STUDENT EDITION Lesson Summary Sometimes we have a scale drawing of something and we want to create another scale drawing of it that uses a different scale We can use the original scale drawing to find the size of the actual object Then we can use the size of the actual object to figure out the size of our new scale drawing For example here is a scale drawing of a park where the scale is 1 cm to 90 m The rectangle is 10 cm by 5 5 cm so the actual dimensions of the park are 900 m by 495 m because 10 90 900 and 5 5 90 495 Suppose we want to make another scale drawing of the park where the scale is 1 cm to 30 meters This new scale drawing should be 30 cm by 16 5 cm because 900 30 30 and 495 30 16 5 Another way to find this answer is to think about how the two different scales are related to each other In the first scale drawing 1 cm represented 90 m In the new drawing we would need 3 cm to represent 90 m That means each length in the new scale drawing should be 3 times as long as it was in the original drawing The new scale drawing should be 30 cm by 16 5 cm because 3 10 30 and 3 5 5 16 5 Since the length and width are 3 times as long the area of the new scale drawing will be 9 times as large as the area of the original scale drawing because 32 9 58 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 10 Date GRADE 7 MISSION 1 LESSON 10 Exit Ticket Here is a scale drawing of a window frame that uses a scale of 1 cm to 6 inches Create another scale drawing of the window frame that uses a scale of 1 cm to 12 inches 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 59

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ZEARN MATH STUDENT EDITION G7M1 LESSON 11 Lesson 11 Scales Without Units Let s explore a different way to express scales Warm Up 1 A map of a park says its scale is 1 to 100 1 What do you think that means 2 Give an example of how this scale could tell us about measurements in the park Concept Exploration ACTIVITY 1 2 Your teacher will give you a drawing of the Apollo Lunar Module It is drawn at a scale of 1 to 50 1 The legs of the spacecraft are its landing gear Use the drawing to estimate the actual length of each leg on the sides Write your answer to the nearest 10 centimeters Explain or show your reasoning 2 Use the drawing to estimate the actual height of the Apollo Lunar Module to the nearest 10 centimeters 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 61

G7M1 LESSON 11 ZEARN MATH STUDENT EDITION 3 Neil Armstrong was 71 inches tall when he went to the surface of the moon in the Apollo Lunar Module How tall would he be in the drawing if he were drawn with his height to scale Show your reasoning 4 Sketch a stick figure to represent yourself standing next to the Apollo Lunar Module Make sure the height of your stick figure is to scale Show how you determined your height on the drawing ACTIVITY 2 3 62 A rectangular parking lot is 120 feet long and 75 feet wide Lin and Diego both made scale drawings of the parking lot Lin made a scale drawing of the parking lot at a scale of 1 inch to 15 feet The drawing she produced is 8 inches by 5 inches Diego made another scale drawing of the parking lot at a scale of 1 to 180 The drawing he produced is also 8 inches by 5 inches 1 Explain or show how each scale would produce an 8 inch by 5 inch drawing 2 Make another scale drawing of the same parking lot at a scale of 1 inch to 20 feet Be prepared to explain your reasoning 3 Express the scale of 1 inch to 20 feet as a scale without units 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 G7M1 LESSON 11 Lesson Summary In some scale drawings the scale specifies one unit for the distances on the drawing and a different unit for the actual distances represented For example a drawing could have a scale of 1 cm to 10 km In other scale drawings the scale does not specify any units at all For example a map may simply say the scale is 1 to 1 000 In this case the units for the scaled measurements and actual measurements can be any unit so long as the same unit is being used for both So if a map of a park has a scale 1 to 1 000 then 1 inch on the map represents 1 000 inches in the park and 12 centimeters on the map represent 12 000 centimeters in the park In other words 1 000 is the scale factor that relates distances 1 on the drawing to actual distances and 1 000 is the scale factor that relates an actual distance to its corresponding distance on the drawing A scale with units can be expressed as a scale without units by converting one measurement in the scale into the same unit as the other usually the unit used in the drawing For example these scales are equivalent 1 inch to 200 feet 1 inch to 2 400 inches because there are 12 inches in 1 foot and 200 12 2 400 1 to 2 400 This scale tells us that all actual distances are 2 400 times their corresponding distances on the drawing 1 and distances on the drawing are 2 400 times the actual distances they represent 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 Name G7M1 LESSON 11 Date GRADE 7 MISSION 1 LESSON 11 Exit Ticket Andre drew a plan of a courtyard at a scale of 1 to 60 On his drawing one side of the courtyard is 2 75 inches 1 What is the actual measurement of that side of the courtyard Express your answer in inches and then in feet 2 If Andre made another courtyard scale drawing at a scale of 1 to 12 would this drawing be smaller or larger than the first drawing 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 65

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ZEARN MATH STUDENT EDITION G7M1 LESSON 12 Lesson 12 Units in Scale Drawings Let s use different scales to describe the same drawing Warm Up 1 There are 2 54 cm in an inch 12 inches in a foot and 5 280 feet in a mile Which expression gives the number of centimeters in a mile Explain your reasoning a 2 54 12 5 280 b 5 280 12 2 54 c 1 5 280 12 2 54 d 5 280 12 2 54 e 5 280 12 2 54 Concept Exploration ACTIVITY 1 2 1 As of 2016 Tunisia holds the world record for the largest version of a national flag It was almost as long as four soccer fields The flag has a circle in the center a crescent moon inside the circle and a star inside the crescent moon Complete the table Explain or show your reasoning Flag Length Actual At 1 to 2 000 scale Flag Height 396 m Height of crescent moon 99 m 13 2 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 67

G7M1 LESSON 12 2 ZEARN MATH STUDENT EDITION Complete each scale with the value that makes it equivalent to the scale of 1 to 2 000 Explain or show your reasoning a 1 cm to cm b 1 cm to m c km 1 cm to d 2 m to e 5 cm to m m f cm to 1 000 m g mm to 20 m 3 a What is the area of the large flag b What is the area of the smaller flag c 68 The area of the large flag is how many times the area of the smaller flag 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M1 LESSON 12 Lesson Summary Sometimes scales come with units and sometimes they don t For example a map of Nebraska may have a scale of 1 mm to 1 km This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska The same scale without units is 1 to 1 000 000 which means that each unit of distance on the map represents 1 000 000 units of distance in Nebraska This is true for any choice of unit To see that these two scales are equivalent notice that there are 1 000 millimeters in 1 meter and 1 000 meters in 1 kilometer This means there are 1 000 1 000 or 1 000 000 millimeters in 1 kilometer So the actual distances in Nebraska are 1 000 000 times as far as the distances on the map A scale tells us how a length on a drawing corresponds to an actual length and it also tells us how an area on a drawing corresponds to an actual area For example if 1 centimeter on a scale drawing represents 2 meters in actual distance what does 1 square centimeter on the drawing represent in actual area The square on the left shows a square with side lengths 1 cm so its area is 1 square cm 1 cm 2m 1 cm 2m The square on the right shows the actual dimensions represented by the square on the left Because each side length in the actual square is 2 m the actual square has an area of 22 or 4 square meters We can use this relationship to find the actual area of any region represented on this drawing If a room has an area of 18 cm2 on the drawing we know that it has an actual area of 18 4 72 or 72 m2 In general if 1 unit on the drawing represents n actual units then one square unit on the drawing represents n2 actual square units 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 69

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ZEARN MATH STUDENT EDITION Name G7M1 LESSON 12 Date GRADE 7 MISSION 1 LESSON 12 Exit Ticket Lin and her brother each created a scale drawing of their backyard but at different scales Lin used a scale of 1 inch to 1 foot Her brother used a scale of 1 inch to 1 yard 1 Express the scales for the drawings without units 2 Whose drawing is larger How many times as large is it 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 71

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ZEARN MATH STUDENT EDITION G7M1 LESSON 13 Lesson 13 Draw It to Scale Let s draw a floor plan Concept Exploration ACTIVITY 1 1 On a blank sheet of paper make a rough sketch of a floor plan of the classroom 1 Include parts of the room that the class has decided to include or that you would like to include Accuracy is not important for this rough sketch but be careful not to omit important features like a door 2 Trade sketches with a partner and check each other s work Specifically check if any parts are missing or incorrectly placed Return their work and revise your sketch as needed 3 Discuss with your group a plan for measuring Work to reach an agreement on a Which classroom features must be measured and which are optional b The units to be used c How to record and organize the measurements on the sketch in a list in a table etc d How to share the measuring and recording work or the role each group member will play 4 Gather your tools take your measurements and record them as planned Be sure to double check your measurements 5 Make your own copy of all the measurements that your group has gathered if you haven t already done so You will need them for the next activity 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M1 LESSON 13 ZEARN MATH STUDENT EDITION ACTIVITY 2 2 1 Determine an appropriate scale for your drawing based on your measurements and your paper choice Your floor plan should fit on the paper and not end up too small Use the scale and the measurements your group has taken to draw a scale floor plan of the classroom Make sure to a Show the scale of your drawing b Label the key parts of your drawing the walls main openings etc with their actual measurements c Show your thinking and organize it so it can be followed by others ACTIVITY 3 3 74 Compare floor plans with other students then record ideas for how your floor plan could be improved 1 Trade floor plans with another student who used the same paper size as you Discuss your observations and thinking 2 Trade floor plans with another student who used a different paper size than you Discuss your observations and thinking 3 Based on your discussions name some ways your floor plan could be improved 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Grade 7 Mission 2 Introducing Proportional Relationships

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ZEARN MATH STUDENT EDITION G7M2 LESSON 1 Lesson 1 One of These Things Is Not Like the Others Let s remember what equivalent ratios are Warm Up 1 1 Use the double number line to answer the following questions Complete the double number line diagram with the missing numbers 0 2 0 1 7 10 5 2 What could each of the number lines represent Invent a situation and label the diagram 3 Make sure your labels include appropriate units of measure Concept Exploration ACTIVITY 1 2 1 There are three mixtures Two taste the same and one is different Answer the questions Which mixture tastes different Describe how it 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 77

G7M2 LESSON 1 2 ZEARN MATH STUDENT EDITION Here are the recipes that were used to make the three mixtures 1 cup of water with 1 2 cups of water with 1 2 1 2 1 cup of water with teaspoons of powdered drink mix teaspoon of powdered drink mix 1 4 teaspoon of powdered drink mix Which of these recipes is for the stronger tasting mixture Explain how you know ACTIVITY 2 3 Here are four different crescent moon shapes Use the shapes to answer the questions B A D C 78 1 What do moons A B and C all have in common that moon D doesn t 2 Use numbers to describe how moons A B and C are different from moon 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 3 G7M2 LESSON 1 Use a table or a double number line to show how moons A B and C are different from moon D Lesson Summary When two different situations can be described by equivalent ratios that means they are alike in some important way An example is a recipe If two people make something to eat or drink the taste will only be the same as long as the ratios of the ingredients are equivalent For example all of the mixtures of water and drink mix in this table taste the same because the ratios of cups of water to scoops of drink mix are all equivalent ratios If a mixture were not equivalent to these for example if the ratio of cups of water to scoops of drink mix were 6 4 then the mixture would taste different Water Cups Drink Mix Scoops 3 1 12 4 1 5 0 5 Notice that the ratios of pairs of corresponding side lengths are equivalent in figures A B and C For example the ratios of the length of the top side to the length of the left side for figures A B and C are equivalent ratios Figures A B and C are scaled copies of each other this is the important way in which they are alike 3 2 A 1 5 4 5 B 3 C 3 3 D If a figure has corresponding sides that are not in a ratio equivalent to these like figure D then it s not a scaled copy In this mission you will study relationships like these that can be described by a set of equivalent ratios TERMINOLOGY Equivalent ratios 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 79

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ZEARN MATH STUDENT EDITION Name G7M2 LESSON 1 Date GRADE 7 MISSION 2 LESSON 1 Exit Ticket Here are three different recipes for Orangey Pineapple Juice Two of these mixtures taste the same and one tastes different Recipe 1 Mix 4 cups of orange juice with 6 cups of pineapple juice Recipe 2 Mix 6 cups of orange juice with 9 cups of pineapple juice Recipe 3 Mix 9 cups of orange juice with 12 cups of pineapple juice Which two recipes will taste the same and which one will taste different 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 81

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ZEARN MATH STUDENT EDITION G7M2 LESSON 2 Lesson 2 Introducing Proportional Relationships with Tables Let s solve problems involving proportional relationships using tables Warm Up 1 Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases What do you notice about the table What do you wonder Number of cases they order Number of rolls of paper towels 1 12 3 36 5 60 2 2 10 120 Concept Exploration ACTIVITY 1 2 1 2 A recipe says that 2 cups of dry rice will serve 6 people Complete the table as you answer the questions Be prepared to explain your reasoning How many people will 10 cups of rice serve How many cups of rice are needed to serve 45 people Cups of rice Number of people 2 6 3 9 10 45 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 2 3 ZEARN MATH STUDENT EDITION A recipe says that 6 spring rolls will serve 3 people Complete the table Number of spring rolls Number of people 6 3 30 40 28 ACTIVITY 2 43 A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough Complete the table as you answer the questions Be prepared to explain your reasoning Note Some days they bake bigger batches and some days they bake smaller batches but they always use the same ratio of honey to flour 1 2 84 How many cups of flour do they use with 20 tablespoons of honey How many cups of flour do they use with 13 tablespoons of honey 3 How many tablespoons of honey do they use with 20 cups of flour 4 What is the proportional relationship represented by this table Honey tbsp Flour c 8 10 20 13 20 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 2 Lesson Summary If the ratios between two corresponding quantities are always equivalent the relationship between the quantities is called a proportional relationship This table shows different amounts of milk and chocolate syrup The ingredients in each row when mixed together would make a different total amount of chocolate milk but these mixtures would all taste the same Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to 4 1 About the relationship between these quantities we could say The relationship between amount of chocolate syrup and amount of milk is proportional The relationship between the amount of chocolate syrup and the amount of milk is a proportional relationship The table represents a proportional relationship between the amount of chocolate syrup and amount of milk The amount of milk is proportional to the amount of chocolate syrup We could multiply any value in the chocolate syrup column by 14 to get the value in the milk column We might call 14 a unit rate because 14 cups of milk are needed for 1 tablespoon of chocolate syrup We also say that 14 is the constant of proportionality for this relationship It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup TERMINOLOGY Constant of proportionality Proportional relationship 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 2 Name Date GRADE 7 MISSION 2 LESSON 2 Exit Ticket When you mix two colors of paint in equivalent ratios the resulting color is always the same Complete the table as you answer the questions 1 How many cups of yellow paint should you mix with 1 cup of blue paint to make the same shade of green Explain or show your reasoning 2 Make up a new pair of numbers that would make the same shade of green Explain how you know they would make the same shade of green 3 What is the proportional relationship represented by this table 4 What is the constant of proportionality What does it represent Cups of blue paint Cups of yellow paint 2 10 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 87

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ZEARN MATH STUDENT EDITION G7M2 LESSON 3 Lesson 3 More about Constant of Proportionality Let s solve more problems involving proportional relationships using tables Warm Up 1 Use the numbers and units from the list to find as many equivalent measurements as you can For example you might write 30 minutes is 12 hour You can use the numbers and units more than once 1 1 2 12 40 0 4 0 3 centimeter 24 meter 1 5 0 01 60 3 50 1 3 30 hour 6 feet 2 minute inch Concept Exploration ACTIVITY 1 2 There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters There are two ways of thinking about this proportional relationship centimeters 0 1 2 3 4 5 0 10 20 30 40 50 millimeters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 3 1 If you know the length of something in centimeters you can calculate its length in millimeters a Complete the table b What is the constant of proportionality ZEARN MATH STUDENT EDITION Length cm Length mm 9 12 5 50 88 49 3 If you know the length of something in millimeters you can calculate its length in centimeters a Complete the table b What is the constant of proportionality Length mm Length cm 70 245 4 699 1 3 How are these two constants of proportionality related to each other 4 Complete each sentence a To convert from centimeters to millimeters you can multiply by b To convert from millimeters to centimeters you can divide by 90 or multiply by 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 3 ACTIVITY 2 3 A plane traveling at a constant speed flew over Pittsburgh Saint Louis Albuquerque and Phoenix on its way from New York to San Diego Complete the table as you answer the questions Be prepared to explain your reasoning Map of the path of a plane flying from New York to San Diego by United States Census Bureau via American Fact Finder Public Domain Segment Time Distance Pittsburgh to Saint Louis 1 hour 550 miles Saint Louis to Albuquerque 1 hour 42 minutes Albuquerque to Phoenix Speed 330 miles 1 What is the distance between Saint Louis and Albuquerque 2 How many minutes did it take to fly between Albuquerque and Phoenix 3 What is the proportional relationship represented by this table 4 Diego says the constant of proportionality is 550 Andre says the constant of proportionality is 9 16 Do you agree with either of them 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 91

G7M2 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary When something is traveling at a constant speed there is a proportional relationship between the time it takes and the distance traveled The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk Distance Traveled cm Elapsed time sec 3 2 1 1 2 3 3 2 10 20 3 2 3 We can multiply any number in the first column by 23 to get the corresponding number in the second column We can say that the elapsed time is proportional to the distance traveled and the constant of proportionality is 23 This means that the bug s pace is 23 seconds per centimeter This table represents the same situation except the columns are switched Distance Traveled cm Elapsed time sec 1 3 2 2 3 1 2 3 20 3 10 3 2 We can multiply any number in the first column by 32 to get the corresponding number in the second column We can say that the distance traveled is proportional to the elapsed time and the constant of 3 3 proportionality is 2 This means that the bug s speed is 2 centimeters per second Notice that 32 is the reciprocal of 23 When two quantities are in a proportional relationship there are two constants of proportionality and they are always reciprocals of each other When we represent a proportional relationship with a table we say the quantity in the second column is proportional to the quantity in the first column and the corresponding constant of proportionality is the number we multiply values in the first column to get the values in the second 92 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 3 Name Date GRADE 7 MISSION 2 LESSON 3 Exit Ticket Mai is filling her fish tank Water flows into the tank at a constant rate Complete the table as you answer the questions Time minutes Water gallons 0 5 0 8 1 3 40 1 How many gallons of water will be in the fish tank after 3 minutes Explain your reasoning 2 How long will it take to fill the tank with 40 gallons of water Explain your reasoning 3 What is the constant of proportionality 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 4 Lesson 4 Proportional Relationships and Equations Let s write equations describing proportional relationships Warm Up 1 Find each quotient mentally 1 645 100 2 645 50 3 48 6 30 4 48 6 x Concept Exploration ACTIVITY 1 2 A recipe says that 2 cups of dry rice will serve 6 people Complete the table as you answer the questions Be prepared to explain your reasoning a How many people will 1 cup of rice serve Cups of Dry Rice Number of People 1 b How many people will 3 cups of rice serve 12 cups 43 cups 2 How many people will x cups of rice serve 12 c 6 3 43 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 95

G7M2 LESSON 4 3 ZEARN MATH STUDENT EDITION A recipe says that 6 spring rolls will serve 3 people Complete the table as you answer the questions Be prepared to explain your reasoning a How many people will 1 spring roll serve Number of Spring Rolls Number of People 1 b How many people will 10 spring rolls serve 16 spring rolls 25 spring rolls c 6 3 10 16 How many people will spring rolls serve 25 n ACTIVITY 2 43 A plane flew at a constant speed between Denver and Chicago It took the plane 1 5 hours to fly 915 miles Complete the table Map of the midwest from Denver to Chicago by United States Census Bureau via American Fact Finder Public Domain Time Hours Distance Miles Speed Miles per Hour 1 1 5 915 2 2 5 t 96 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M2 LESSON 4 1 How far does the plane fly in one hour 2 How far would the plane fly in t hours at this speed 3 If d represents the distance that the plane flies at this speed for t hours write an equation that relates t and d 4 How far would the plane fly in 3 hours at this speed In 3 5 hours 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 97

G7M2 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary The table shows the amount of red paint and blue paint needed to make a certain shade of purple paint called Venusian Sunset Note that parts can be any unit for volume If we mix 3 cups of red with 12 cups of blue you will get the same shade as if we mix 3 teaspoons of red with 12 teaspoons of blue Red Paint Parts Blue Paint Parts 3 12 1 4 7 28 1 4 1 r 4r The last row in the table says that if we know the amount of red paint needed r we can always multiply it by 4 to find the amount of blue paint needed b to mix with it to make Venusian Sunset We can say this more succinctly with the equation b 4r So the amount of blue paint is proportional to the amount of red paint and the constant of proportionality is 4 We can also look at this relationship the other way around If we know the amount of blue paint needed b we can always multiply it by 14 to find the amount of red paint needed r to mix with it to make Venusian Sunset So r 14 b The amount of blue paint is proportional to the amount of red paint and the constant of proportionality 14 Red Paint Parts Blue Paint Parts 12 3 4 1 28 7 1 1 4 b 1 4 b In general when y is proportional to x we can always multiply x by the same number k the constant of proportionality to get y We can write this much more succinctly with the equation y kx 98 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 4 Date GRADE 7 MISSION 2 LESSON 4 Exit Ticket Snow is falling steadily in Syracuse New York After 2 hours 4 inches of snow has fallen 1 If it continues to snow at the same rate how many inches of snow would you expect after 6 5 hours If you get stuck you can use the table to help Time Hours Snow Inches 1 1 2 2 3 Write an equation that gives the amount of snow that has fallen after x hours at this rate 4 6 5 x How many inches of snow will fall in 24 hours if it continues to snow at this rate 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 5 Lesson 5 Two Equations for Each Relationship Let s investigate the equations that represent proportional relationships Warm Up 1 Here are the second and fourth figures in a pattern Use them to answer the following questions 1 What do you think the first and third figures in the pattern look like 2 Describe the 10th figure in the pattern Concept Exploration ACTIVITY 1 2 1 There are 100 centimeters cm in every meter m Answer the following questions about this relationship Complete each of the tables length m length cm length cm length m 1 100 100 1 0 94 250 1 67 78 2 57 24 123 9 x y 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 5 ZEARN MATH STUDENT EDITION 2 For each table find the constant of proportionality 3 What is the relationship between these constants of proportionality 4 For each table write an equation for the proportional relationship Let x represent a length measured in meters and y represent the same length measured in centimeters ACTIVITY 2 3 1 It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate Let w be the number of gallons of water in the cooler after t minutes Which of the following equations represent the relationship between w and t Select all that apply a w 1 6t b w 0 625t c t 1 6w d t 0 625w 102 2 What does 1 6 tell you about the situation 3 What does 0 625 tell you about the situation 4 Priya changed the rate at which water flowed through the faucet Write an equation that represents the relationship of w and t when it takes 3 minutes to fill the cooler with 1 gallon of water 5 Was the cooler filling faster before or after Priya changed the rate of water flow 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 G7M2 LESSON 5 Lesson Summary If Kiran rode his bike at a constant 10 miles per hour his distance in miles d is proportional to the number of hours t that he rode We can write the equation d 10t With this equation it is easy to find the distance Kiran rode when we know how long it took because we can just multiply the time by 10 We can rewrite the equation d 10t 101 d t 1 t 10 d This version of the equation tells us that the amount of time he rode is proportional to the distance he 1 traveled and the constant of proportionality is 10 That form is easier to use when we know his distance 1 and want to find how long it took because we can just multiply the distance by 10 When two quantities x and y are in a proportional relationship we can write the equation y kx and say y is proportional to x In this case the number k is the corresponding constant of proportionality We can also write the equation x 1 k y and say x is proportional to y In this case the number 1k is the corresponding constant of proportionality Each one can be useful depending on the information we have and the quantity we are trying to figure out 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 103

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ZEARN MATH STUDENT EDITION Name G7M2 LESSON 5 Date GRADE 7 MISSION 2 LESSON 5 Exit Ticket An albatross is a large bird that can fly 400 kilometers in 8 hours at a constant speed Using d for distance in kilometers and t for number of hours an equation that represents this situation is d 50t 1 What are two constants of proportionality for the relationship between distance in kilometers and number of hours What is the relationship between these two values 2 Write another equation that relates d and t in this context 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 105

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ZEARN MATH STUDENT EDITION G7M2 LESSON 6 Lesson 6 Using Equations to Solve Problems Let s use equations to solve problems involving proportional relationships Warm Up 1 Without calculating order the quotients of these expressions from least to greatest a 42 6 0 07 b 42 6 70 c 42 6 0 7 d 426 70 2 a Place the decimal point in the appropriate location in the quotient 42 6 7 608571 b Use this answer to find the quotient of one of the previous expressions Concept Exploration ACTIVITY 1 2 A performer expects to sell 5 000 tickets for an upcoming concert They want to make a total of 311 000 in sales from these tickets 1 Assuming that all tickets have the same price what is the price for one ticket 2 How much will they make if they sell 7 000 tickets 3 How much will they make if they sell 10 000 tickets 50 000 120 000 a million x tickets 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 6 4 If they make 379 420 how many tickets have they sold 5 How many tickets will they have to sell to make 5 000 000 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 108 Aluminum cans can be recycled instead of being thrown in the garbage The weight of 10 aluminum cans is 0 16 kilograms The aluminum in 10 cans that are recycled has a value of 0 14 1 If a family threw away 2 4 kg of aluminum in a month how many cans did they throw away Explain or show your reasoning 2 What would be the recycled value of those same cans Explain or show your reasoning 3 Write an equation to represent the number of cans c given their weight w 4 Write an equation to represent the recycled value r of c cans 5 Write an equation to represent the recycled value r of w kilograms of aluminum 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 6 Lesson Summary Remember that if there is a proportional relationship between two quantities their relationship can be represented by an equation of the form y kx Sometimes writing an equation is the easiest way to solve a problem For example we know that Denali the highest mountain peak in North America is 20 310 feet above sea level How many miles is that There are 5 280 feet in 1 mile This relationship can be represented by the equation f 5 280m where f represents a distance measured in feet and m represents the same distance measured miles Since we know Denali is 20 310 feet above sea level we can write 20 310 5 280m So m 20 310 which is approximately 3 85 miles 5 280 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 109

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ZEARN MATH STUDENT EDITION Name G7M2 LESSON 6 Date GRADE 7 MISSION 2 LESSON 6 Exit Ticket Based on her recipe Elena knows that 5 servings of granola have 1 750 calories 1 If she eats 2 servings of granola how many calories does she eat 2 If she wants to eat 175 calories of granola how many servings should she eat 3 Write an equation to represent the relationship between the number of calories and the number of servings of granola 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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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ZEARN MATH STUDENT EDITION G7M2 LESSON 7 Lesson 7 Comparing Relationships with Tables Let s explore how proportional relationships are different from other relationships Warm Up 1 A lemonade recipe calls for the juice of 5 lemons 2 cups of water and 2 tablespoons of honey Invent four new versions of this lemonade recipe 1 One that would make more lemonade but taste the same as the original recipe 2 One that would make less lemonade but taste the same as the original recipe 3 One that would have a stronger lemon taste than the original recipe 4 One that would have a weaker lemon taste than the original recipe 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 113

G7M2 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Entrance to a state park costs 6 per vehicle plus 2 per person in the vehicle How much would it cost for a car with 2 people to enter the park 4 people 10 people Record your answers in the table Number of People in Vehicle Total Entrance Cost in Dollars 2 4 10 114 2 For each row in the table if each person in the vehicle splits the entrance cost equally how much will each person pay 3 How might you determine the entrance cost for a bus with 50 people 4 Is the relationship between the number of people and the total entrance cost a proportional relationship 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 G7M2 LESSON 7 ACTIVITY 2 3 Han and Clare were running laps around the track The coach recorded their times at the end of laps 2 4 6 and 8 Han s run Distance laps Time minutes 2 Clare s run Minutes per Lap Distance laps Time minutes 4 2 5 4 9 4 10 6 15 6 15 8 23 8 20 Minutes per Lap 1 Is Han running at a constant pace Is Clare How do you know 2 Write an equation for the relationship between distance and time for anyone who is running at a constant pace 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary Here are the prices for some smoothies at two different smoothie shops Smoothie Shop A Smoothie Shop B Smoothie Size oz Price Dollars Per Ounce Smoothie Size oz Price Dollars Per Ounce 8 6 0 75 8 6 0 75 12 9 0 75 12 8 0 67 16 12 0 75 16 10 0 625 s 0 75s 0 75 s For Smoothie Shop A smoothies cost 0 75 per ounce no matter which size we buy There could be a proportional relationship between smoothie size and the price of the smoothie An equation representing this relationship is p 0 75 s where s represents size in ounces and p represents price in dollars The relationship could still not be proportional if there were a different size on the menu that did not have the same price per ounce For Smoothie Shop B the cost per ounce is different for each size Here the relationship between smoothie size and price is definitely not proportional In general two quantities in a proportional relationship will always have the same quotient When we see some values for two related quantities in a table and we get the same quotient when we divide them that means they might be in a proportional relationship but if we can t see all of the possible pairs we can t be completely sure However if we know the relationship can be represented by an equation is of the form y kx then we are sure it is proportional 116 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 7 Name Date GRADE 7 MISSION 2 LESSON 7 Exit Ticket 1 2 3 Based on the information in the table is the cost of the apples proportional to the weight of apples Pounds of Apples Cost of Apples 2 3 76 3 5 64 4 7 52 5 9 40 Based on the information in the table is the cost of the pizza proportional to the number of toppings Number of Toppings Cost of Pizza 2 11 99 3 13 49 4 14 99 5 16 49 Write an equation for the proportional relationship 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 117

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ZEARN MATH STUDENT EDITION G7M2 LESSON 8 Lesson 8 Comparing Relationships with Equations Let s develop methods for deciding if a relationship is proportional Warm Up 1 Do you see a pattern What predictions can you make about future rectangles in the set if your pattern continues Concept Exploration ACTIVITY 1 2 1 The other day you worked with converting meters centimeters and millimeters Here are some more unit conversions Use the equation F 95 C 32 where F represents degrees Fahrenheit and C represents degrees Celsius to complete the table Temperature 0C Temperature 0F 20 4 175 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 8 2 ZEARN MATH STUDENT EDITION Use the equation c 2 54n where c represents the length in centimeters and n represents the length in inches to complete the table Length in Length cm 10 8 3 3 1 2 Are these proportional relationships Explain why or why not ACTIVITY 2 3 Here are some cubes with different side lengths Complete each table and answer the questions Be prepared to explain your reasoning 3 1 How long is the total edge length of each cube Side length 2 1 2 What is the surface area of each cube Side length 3 3 5 5 9 1 2 s 120 Total edge length 9 5 9 Surface area 1 2 s 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 G7M2 LESSON 8 What is the volume of each cube Side length Volume 3 5 9 1 2 s 4 Which of these relationships is proportional Explain how you know 5 Write equations for the total edge length E total surface area A and volume V of a cube with side length s 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 121

G7M2 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary If two quantities are in a proportional relationship then their quotient is always the same This table represents different values of a and b two quantities that are in a proportional relationship a b b a 20 100 5 3 15 5 11 55 5 1 5 5 Notice that the quotient of b and a is always 5 To write this as an equation we could say true then b 5a This doesn t work if a 0 but it works otherwise b a 5 If this is y If quantity y is proportional to quantity x we will always see this pattern x will always have the same value This value is the constant of proportionality which we often refer to as k We can represent this y relationship with the equation x k as long as x is not 0 or y kx Note that if an equation cannot be written in this form then it does not represent a proportional relationship 122 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 8 Date GRADE 7 MISSION 2 LESSON 8 Exit Ticket Andre is setting up rectangular tables for a party He can fit 6 chairs around a single table Andre lines up 10 tables end to end and tries to fit 60 chairs around them but he is surprised when he cannot fit them all 1 Write an equation for the relationship between the number of chairs c and the number of tables t when a the tables are apart from each other b the tables are placed end to end 3 Is the first relationship proportional Explain how you know 4 Is the second relationship proportional 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 123

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ZEARN MATH STUDENT EDITION G7M2 LESSON 9 Lesson 9 Solving Problems about Proportional Relationships Let s solve problems about proportional relationships Warm Up 1 Consider the problem A person is running a distance race at a constant rate What time will they finish the race What information would you need to be able to solve the problem 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 Follow the direction below If your teacher gives you the problem card 1 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 If your teacher gives you the data card 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 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 125

G7M2 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary Whenever we have a situation involving constant rates we are likely to have a proportional relationship between quantities of interest When a bird is flying at a constant speed then there is a proportional relationship between the flying time and distance flown If water is filling a tub at a constant rate then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up If an aardvark is eating termites at a constant rate then there is proportional relationship between the number of termites the aardvark has eaten and the time since it started eating Sometimes we are presented with a situation and it is not so clear whether a proportional relationship is a good model How can we decide if a proportional relationship is a good representation of a particular situation If you aren t sure where to start look at the quotients of corresponding values If they are not always the same then the relationship is definitely not a proportional relationship If you can see that there is a single value that we always multiply one quantity by to get the other quantity it is definitely a proportional relationship After establishing that it is a proportional relationship setting up an equation is often the most efficient way to solve problems related to the situation 126 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 9 Date GRADE 7 MISSION 2 LESSON 9 Exit Ticket A steel beam can be cut to different lengths for a project Assuming the weight of a steel beam is proportional to its length what information would you need to know to write an equation that represents this relationship 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M2 LESSON 10 Lesson 10 Introducing Graphs of Proportional Relationships Let s see how graphs of proportional relationships differ from graphs of other relationships Warm Up 1 1 Plot the points and answer the question Plot the points 0 10 1 8 2 6 3 4 4 2 y 11 10 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 11 x 1 2 What do you notice about the graph 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 129

G7M2 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Some T shirts cost 8 each Use the table to answer the questions Use the table to answer these questions a What does x represent b What does y represent c 2 Is there a proportional relationship between x and y Plot the pairs in the table on the coordinate plane y x y 1 8 2 16 3 24 4 32 5 40 6 48 50 40 30 20 10 1 3 2 3 4 5 6 x What do you notice about the graph ACTIVITY 2 3 Use the tables and graphs you receive to answer the questions Your teacher will give you papers showing tables and graphs 1 130 Examine the graphs closely What is the same and what is different about the graphs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 10 2 Sort the graphs into categories of your choosing Label each category Be prepared to explain why you sorted the graphs the way you did 3 Take turns with a partner to match a table with a graph a For each match you find explain to your partner how you know it is a match b For each match your partner finds listen carefully to their explanation If you disagree work to reach an agreement Pause here so your teacher can review your work 4 Trade places with another group How are their categories the same as your group s categories How are they different 5 Return to your original place Discuss any changes you may wish to make to your categories based on what the other group did 6 Which of the relationships are proportional 7 What have you noticed about the graphs of proportional relationships Do you think this will hold true for all graphs of proportional relationships 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M2 LESSON 10 ZEARN MATH STUDENT EDITION Lesson Summary One way to represent a proportional relationship is with a graph Here is a graph that represents different amounts that fit the situation Blueberries cost 6 per pound Different points on the graph tell us for example that 2 pounds of blueberries cost 12 and 4 5 pounds of blueberries cost 27 Cost in dollars 40 30 20 3 18 2 12 10 Sometimes it makes sense to connect the points with a line and sometimes it doesn t We could buy for example 4 5 pounds of blueberries or 1 875 pounds of blueberries so all the points in between the whole numbers make sense in the situation so any point on the line is meaningful 4 5 27 1 6 1 3 2 4 5 Weight in pounds If the graph represented the cost for different numbers of sandwiches instead of pounds of blueberries it might not make sense to connect the points with a line because it is often not possible to buy 4 5 sandwiches or 1 875 sandwiches Even if only points make sense in the situation though sometimes we connect them with a line anyway to make the relationship easier to see Graphs that represent proportional relationships all have a few things in common Points that satisfy the relationship lie on a straight line The line that they lie on passes through the origin 0 0 Here are some graphs that do not represent proportional relationships y y 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 These points do not lie on a line 5 6 7 x 1 2 3 4 5 6 7 x This is a line but it doesn t go through the origin TERMINOLOGY Origin 132 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M2 LESSON 10 Name Date GRADE 7 MISSION 2 LESSON 10 Exit Ticket Which graphs cannot represent a proportional relationship Select all that apply Explain how you know A B y 30 30 20 20 10 10 2 C y 4 x 5 D y y 60 30 40 20 20 10 2 4 x x 10 5 10 15 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 133

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Grade 7 Mission 3 Measuring Circles

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ZEARN MATH STUDENT EDITION G7M3 LESSON 1 Lesson 1 How Well Can You Measure Let s see how accurately we can measure Warm Up 1 A student got 16 out of 21 questions correct on a quiz Use mental estimation to answer these questions 1 Did the student answer less than or more than 80 of the questions correctly 2 Did the student answer less than or more than 75 of the questions correctly Concept Exploration ACTIVITY 1 2 1 Here are nine squares Your teacher will assign your group three of these squares to examine more closely For each of your assigned squares measure the length of the diagonal and the perimeter of the square in centimeters Diagonal cm Perimeter cm Square A Square B Square C Square D Square E Square F Square G Square H Square I Check your measurements with your group After you come to an agreement record your measurements in the 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 137

G7M3 LESSON 1 2 ZEARN MATH STUDENT EDITION Plot the diagonal and perimeter values from the table on the coordinate plane 40 35 Perimeter cm 30 25 20 15 10 5 5 3 10 Diagonal cm 15 What do you notice about the points on the graph ACTIVITY 2 3 In the table record the length of the diagonal for each of your assigned squares from the previous activity Next calculate the area of each of your squares 1 Diagonal cm Area cm2 Square A Square B Square C Square D Square E Square F Square G Square H Square I Pause here so your teacher can review your work Be prepared to share your values with the class 138 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 1 2 Examine the class graph of these values What do you notice 3 How is the relationship between the diagonal and area of a square the same as the relationship between the diagonal and perimeter of a square from the previous activity How is it 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 139

G7M3 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary When we measure the values for two related quantities plotting the measurements in the coordinate plane can help us decide if it makes sense to model them with a proportional relationship If the points are close to a line through 0 0 then a proportional relationship is a good model For example here is a graph of the values for the height measured in millimeters of different numbers of pennies placed in a stack y Because the points are close to a line through 0 0 the height of the stack of pennies appears to be proportional to the number of pennies in a stack This makes sense because we can see that the heights of the pennies only vary a little bit Height of stack in millimeters 45 40 35 30 25 20 15 10 5 5 10 15 20 25 30 x Number of pennies An additional way to investigate whether or not a relationship is proportional is by making a table Here is some data for the weight of different numbers of pennies in grams along with the corresponding number of grams per penny Number of pennies Grams Grams per penny 1 3 1 3 1 2 5 6 2 8 5 13 1 2 6 10 25 6 2 6 Though we might expect this relationship to be proportional the quotients are not very close to one another In fact the metal in pennies changed in 1982 and older pennies are heavier This explains why the weight per penny for different numbers of pennies are so different 140 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 1 Name Date GRADE 7 MISSION 3 LESSON 1 Exit Ticket 1 The graph shows the height of a plant after a certain amount of time measured in days Plant height cm 25 20 15 10 5 0 Height of snow inches 2 Do you think that there may be a proportional relationship between the number of days and the height of the plant Explain your reasoning 10 20 30 40 Time days 50 60 The graph shows how much snow fell after a certain amount of time measured in hours 4 Do you think that there may be a proportional relationship between the number of hours and the amount of snow that fell Explain your reasoning 3 2 1 0 1 2 3 4 Time hours 5 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 141

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ZEARN MATH STUDENT EDITION G7M3 LESSON 2 Lesson 2 Exploring Circles Let s explore circles Warm Up 1 Here are two figures Figure C looks more like Figure A than like Figure B Sketch what Figure C might look like Explain your reasoning 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 143

G7M3 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 144 Your teacher will give you some pictures of different objects 1 How could you sort these pictures into two groups Be prepared to share your reasoning 2 Work with your partner to sort the pictures into the categories that your class has agreed on Pause here so your teacher can review your work 3 What are some characteristics that all circles have in common 4 Put the circular objects in order from smallest to largest 5 Select one of the pictures of a circular object What are some ways you could measure the actual size of your circle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 2 ACTIVITY 2 3 Priya Han and Mai each measured one of the circular objects from earlier Priya says that the bike wheel is 24 inches Han says that the yo yo trick is 24 inches Mai says that the glow necklace is 24 inches 1 Do you think that all these circles are the same size 2 What part of the circle did each person measure 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 145

G7M3 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary A circle consists of all of the points that are the same distance away from a particular point called the center of the circle A segment that connects the center with any point on the circle is called a radius For example segments QG QH QI and QJ are all radii of circle 2 We say one radius and two radii The length of any radius is always the same for a given circle For this reason people also refer to this distance as the radius of the circle circle 2 circle 1 H I C A P E F Q J B D G A segment that connects two opposite points on a circle passing through the circle s center is called a diameter For example segments AB CD and EF are all diameters of circle 1 All diameters in a given circle have the same length because they are composed of two radii For this reason people also refer to the length of such a segment as the diameter of the circle The circumference of a circle is the distance around it If a circle was made of a piece of string and we cut it and straightened it out the circumference would be the length of that string A circle always encloses a circular region The region enclosed by circle 2 is shaded but the region enclosed by circle 1 is not When we refer to the area of a circle we mean the area of the enclosed circular region TERMINOLOGY Circle Circumference Diameter Radius 146 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M3 LESSON 2 Name Date GRADE 7 MISSION 3 LESSON 2 Exit Ticket Here are two circles Their centers are A and F C H 4 cm E A B F 8 cm D G 1 What is the same about the two circles What is different 2 What is the length of segment AD How do you know 3 On the first circle what segment is a diameter How long is 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 147

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ZEARN MATH STUDENT EDITION G7M3 LESSON 3 Lesson 3 Exploring Circumference Let s explore the circumference of circles Warm Up 1 Clare wonders if the height of the toilet paper tube or the distance around the tube is greater What information would she need in order to solve the problem How could she find this out Concept Exploration ACTIVITY 1 2 1 Your teacher will give you several circular objects Use them to answer the following questions Measure the diameter and the circumference of the circle in each object to the nearest tenth of a centimeter Record your measurements in the table Object 2 Diameter cm Circumference cm Plot the diameter and circumference values from the table on the coordinate plane What do you notice 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M3 LESSON 3 ZEARN MATH STUDENT EDITION y 60 Circumceference cm 50 40 30 20 10 5 0 10 15 20 25 30 x Diameter cm 3 Plot the points from two other groups on the same coordinate plane Do you see the same pattern that you noticed earlier ACTIVITY 2 3 Here are five circles One measurement for each circle is given in the table Use the constant of proportionality estimated in the previous activity to complete the table A C E B D Diameter cm Circle A 3 Circle B 10 Circle C 24 Circle D 18 Circle E 150 Circumference cm 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 G7M3 LESSON 3 Lesson Summary There is a proportional relationship between the diameter and circumference of any circle That means that if we write C for circumference and d for diameter we know that C kd where k is the constant of proportionality The exact value for the constant of proportionality is called Some frequently used approximations for 22 are 7 3 14 and 3 14159 but none of these is exactly We can use this to estimate the circumference if we know the diameter and vice versa For example using 3 1 as an approximation for if a circle has a diameter of 4 cm then the circumference is about 3 1 4 12 4 or 12 4 cm C 12 10 8 6 4 1 2 1 2 3 4 5 6 d The relationship between the circumference and the diameter can be written as 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 151

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ZEARN MATH STUDENT EDITION Name G7M3 LESSON 3 Date GRADE 7 MISSION 3 LESSON 3 Exit Ticket Select all the pairs that could be reasonable approximations for the diameter and circumference of a circle Explain your reasoning a 5 meters and 22 meters b 19 inches and 60 inches c 33 centimeters and 80 centimeters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 153

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ZEARN MATH STUDENT EDITION G7M3 LESSON 4 Lesson 4 Applying Circumference Let s use to solve problems Warm Up 1 Answer the questions about the circular objects below Here are some pictures of circular objects with measurement tools shown The measurement tool on each picture reads as follows Wagon wheel 3 feet Plane propeller 24 inches Sliced Orange 20 centimeters 1 For each picture which measurement is shown 2 Based on this information what measurement s could you estimate for each picture Concept Exploration ACTIVITY 1 2 1 Your teacher will assign you an approximation for to use for this activity Use that approximation to answer the following questions Complete the table Object Radius Wagon wheel Airplane propeller Orange slice Diameter Circumference 3 ft 24 in 20 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 155

G7M3 LESSON 4 2 ZEARN MATH STUDENT EDITION A bug was sitting on the tip of a wind turbine blade that was 24 inches long when it started to rotate The bug held on for 5 rotations before flying away How far did the bug travel before it flew off ACTIVITY 2 3 156 Kiran bent some wire around a rectangle to make a picture frame The rectangle is 8 inches by 10 inches 1 Find the perimeter of the wire picture frame Explain or show your reasoning 2 If the wire picture frame were stretched out to make one complete circle what would its radius be 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 4 Lesson Summary The circumference of a circle C is times the diameter d The diameter is twice the radius r So if we know any one of these measurements for a particular circle we can find the others We can write the relationships between these different measures using equations d 2r C d C 2 r If the diameter of a car tire is 60 cm that means the radius is 30 cm and the circumference is 60 or about 188 cm If the radius of a clock is 5 in that means the diameter is 10 in and the circumference is 10 or about 31 in If a ring has a circumference of 44 mm that means the diameter is 44 which is about 14 mm and the radius is about 7 mm 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION Name G7M3 LESSON 4 Date GRADE 7 MISSION 3 LESSON 4 Exit Ticket Circle A has a diameter of 9 cm Circle B has a radius of 5 cm 1 Which circle has the larger circumference 2 About how many centimeters larger is 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 159

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ZEARN MATH STUDENT EDITION G7M3 LESSON 5 Lesson 5 Circumference and Wheels Let s explore how far different wheels roll Warm Up 1 Han says that you can wrap a 5 foot rope around a wheel with a 2 foot diameter because 52 is less than pi Do you agree with Han Explain your reasoning Lesson ACTIVITY 1 2 1 Your teacher will give you a circular object Follow these instructions to create the drawing a Use a ruler to draw a line all the way across the page If your object is not very small you may want to use a separate piece of paper b Roll your object along the line and mark where it completes one rotation c 2 Use your object to draw tick marks along the line that are spaced as far apart as the diameter of your object What do you notice 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M3 LESSON 5 3 ZEARN MATH STUDENT EDITION Use your ruler to measure each distance Record these values in the first row of the table a the diameter of your object b how far your object rolled in one complete rotation c the quotient of how far your object rolled divided by the diameter of your object Object Diameter Distance traveled in one rotation Distance Diameter 4 If you wanted to trace two complete rotations of your object how long of a line would you need 5 Share your results with your group and record their measurements in the table 6 If each person in your group rolled their object along the entire length of the classroom which object would complete the most rotations Explain or show your reasoning ACTIVITY 2 3 1 Use the information about a car wheel and bike wheel to answer the questions below A car wheel has a diameter of 20 8 inches a About how far does the car wheel travel in 1 rotation 5 rotations 30 rotations 162 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 5 b Write an equation relating the distance the car travels in inches c to the number of wheel rotations x c 2 About how many rotations does the car wheel make when the car travels 1 mile Explain or show your reasoning A bike wheel has a radius of 13 inches a About how far does the bike wheel travel in 1 rotation 5 rotations 30 rotations b Write an equation relating the distance the bike travels in inches b to the number of wheel rotations x c About how many rotations does the bike wheel make when the bike travels 1 mile Explain or show your reasoning ACTIVITY 3 43 The circumference of a car wheel is about 65 inches 1 If the car wheel rotates once per second how far does the car travel in one minute 2 If the car wheel rotates once per second about how many miles does the car travel in one hour 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 163

G7M3 LESSON 5 ZEARN MATH STUDENT EDITION 3 If the car wheel rotates 5 times per second about how many miles does the car travel in one hour 4 If the car is traveling 65 miles per hour about how many times per second does the wheel rotate Lesson Summary The circumference of a circle is the distance around the circle This is also how far the circle rolls on flat ground in one rotation For example a bicycle wheel with a diameter of 24 inches has a circumference of 24 inches and will roll 24 inches or 2 feet in one complete rotation There is an equation relating the number of rotations of the wheel to the distance it has traveled To see why let s look at a table showing how far the bike travels when the wheel makes different numbers of rotations Number of rotations Distance traveled feet 1 2 2 4 3 6 10 20 50 100 x In the table we see that the relationship between the distance traveled and the number of wheel rotations is a proportional relationship The constant of proportionality is 2 To find the missing value in the last row of the table note that each rotation of the wheel contributes 2 feet of distance traveled So after x rotations the bike will travel 2 x feet If d is the distance in feet traveled when this wheel makes x rotations we have the relationship d 2 x 164 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G7M3 LESSON 5 Date GRADE 7 MISSION 3 LESSON 5 Exit Ticket The wheels on Noah s bike have a circumference of about 5 feet 1 How far does the bike travel as the wheel makes 15 complete rotations 2 How many times do the wheels rotate if Noah rides 40 feet 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M3 LESSON 6 Lesson 6 Estimating Areas Let s estimate the areas of weird shapes Warm Up 1 Find a strategy to make each calculation below mentally 1 599 87 2 254 88 3 99 75 Concept Exploration ACTIVITY 1 2 Below is a floor plan of a house Approximate lengths of the walls are given What is the approximate area of the home including the balcony 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 167

G7M3 LESSON 6 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 168 Estimate the area of Nevada in square miles 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 G7M3 LESSON 6 Lesson Summary We can find the area of some complex polygons by surrounding them with a simple polygon like a rectangle For example this octagon is contained in a rectangle The rectangle is 20 units long and 16 units wide so its area is 320 square units To get the area of the octagon we need to subtract the areas of the four right triangles in the corners These triangles are each 8 units long and 5 units wide so they each have an area of 20 square units The area of the octagon is 320 4 20 or 240 square units We can estimate the area of irregular shapes by approximating them with a polygon and finding the area of the polygon For example here is a satellite picture of Lake Tahoe with some one dimensional measurements around the lake The area of the rectangle is 160 square miles and the area of the triangle is 17 5 square miles for a total of 177 5 square miles We recognize that this is an approximation and not likely the exact area of the lake 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 169

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ZEARN MATH STUDENT EDITION Name G7M3 LESSON 6 Date GRADE 7 MISSION 3 LESSON 6 Exit Ticket Estimate the area of Alberta in square miles 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 171

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ZEARN MATH STUDENT EDITION G7M3 LESSON 7 Lesson 7 Exploring the Area of a Circle Let s investigate the areas of circles Warm Up 1 Decide which figure has the largest area Be prepared to explain your reasoning C A B 2 How would you find or estimate the area of each of the figures using the grid Be prepared to explain your reasoning 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 173

G7M3 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 1 Your teacher will assign your group two circles of different sizes For each circle use the squares on the graph paper to measure the diameter and estimate the area of the circle Record your measurements in the table Diameter cm 2 Estimated area cm2 Plot the values from the table on the class coordinate plane Then plot the class s data points on your coordinate plane 300 Area cm2 250 200 150 100 50 2 4 6 8 10 12 14 16 18 20 22 Diameter cm 3 174 In a previous lesson you graphed the relationship between the diameter and circumference of a circle How is this graph the same How is it 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 G7M3 LESSON 7 Lesson Summary The circumference C of a circle is proportional to the diameter d and we can write this relationship as C d The circumference is also proportional to the radius of the circle and the constant of proportionality is 2 because the diameter is twice as long as the radius However the area of a circle is not proportional to the diameter or the radius The area of a circle with radius r is a little more than 3 times the area of a square with side r so the area of a circle of radius r is approximately 3r2 We saw earlier that the circumference of a circle of radius r is 2 r If we write C for the circumference of a circle this proportional relationship can be written C 2 r The area A of a circle with radius r is approximately 3r2 Unlike the circumference the area is not proportional to the radius because 3r2 cannot be written in the form kr for a number k We will investigate and refine the relationship between the area and the radius of a circle in future lessons TERMINOLOGY Area of a circle 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION Name G7M3 LESSON 7 Date GRADE 7 MISSION 3 LESSON 7 Exit Ticket Circle A has a diameter of approximately 20 inches and an area of approximately 300 in2 Circle B has a diameter of approximately 60 inches Which of these could be the area of Circle B Explain your reasoning a About 100 in2 b About 300 in2 c About 900 in2 d About 2 700 in2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M3 LESSON 8 Lesson 8 Relating Area to Circumference Let s rearrange circles to calculate their areas Warm Up 1 A circular field is set into a square with an 800 m side length Estimate the field s area a About 5 000 m2 b About 50 000 m2 c About 500 000 m2 d About 5 000 000 m2 800 m e About 50 000 000 m2 Concept Exploration ACTIVITY 1 2 You will receive a circular object a marker and two pieces of paper of different colors Follow the instructions to create a visual display 1 Using a thick marker trace your circle in two separate places on the same piece of paper 2 Cut out both circles cutting around the marker line 3 Fold and cut one of the circles into fourths 4 Arrange the fourths so that straight sides are next to each other but the curved edges are alternately on top and on bottom Pause here so your teacher can review your work 5 Fold and cut the fourths in half to make eighths Arrange the eighths next to each other like you did with the fourths 6 If your pieces are still large enough repeat the previous step to make sixteenths 7 Glue the remaining circle and the new shape onto a piece of paper that is a different 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 179

G7M3 LESSON 8 ZEARN MATH STUDENT EDITION After you finish gluing your shapes answer the following questions 1 How do the areas of the two shapes compare 2 What polygon does the shape made of the circle pieces most resemble 3 How could you find the area of this polygon ACTIVITY 2 3 Elena wants to tile the top of a circular table The diameter of the table top is 28 inches What is its area Lesson Summary If C is a circle s circumference and r is its radius then C 2 r The area of a circle can be found by taking the product of half the circumference and the radius 1 2 If is the area of the circle this gives the equation A 2 r r This equation can be rewritten as A r2 This means that if we know the radius we can find the area For example if a circle has radius 10 cm then the area is about 3 14 100 which is 314 cm2 If we know the diameter we can figure out the radius and then we can find the area For example if a circle has a diameter of 30 ft then the radius is 15 ft and the area is about 3 14 225 which is approximately 707 ft2 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

ZEARN MATH STUDENT EDITION G7M3 LESSON 8 Name Date GRADE 7 MISSION 3 LESSON 8 Exit Ticket A circle s circumference is approximately 44 cm Complete each statement using one of these values 7 11 14 22 88 138 154 196 380 616 1 The circle s diameter is approximately 2 The circle s radius is approximately 3 The circle s area is approximately cm cm cm2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M3 LESSON 9 Lesson 9 Applying Area of Circles Let s find the areas of shapes made up of circles Warm Up 1 The area of this field is about 500 000 m2 What is the field s area to the nearest square meter Assume that the side lengths of the square are exactly 800 m 800 m a 502 400 m2 b 502 640 m2 c 502 655 m2 d 502 656 m2 e 502 857 m2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M3 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Each square has a side length of 12 units Compare the areas of the shaded regions in the 3 figures Which figure has the largest shaded region Explain or show your reasoning A 3 B Each square in Figures D and E has a side length of 1 unit Compare the area of the two figures Which figure has more area How much more Explain or show your reasoning D 184 C 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 G7M3 LESSON 9 Lesson Summary The relationship between A the area of a circle and r its radius is A r2 We can use this to find the area of a circle if we know the radius For example if a circle has a radius of 10 cm then the area is 102 or 100 cm2 We can also use the formula to find the radius of a circle if we know the area For example if a circle has an area of 49 m2 then its radius is 7 m and its diameter is 14 m Sometimes instead of leaving in expressions for the area a numerical approximation can be helpful For the examples above a circle of radius 10 cm has area about 314 cm2 In a similar way a circle with area 154 m2 has radius about 7 m We can also figure out the area of a fraction of a circle For example the figure shows a circle divided 1 into 3 pieces of equal area The shaded part has an area of 3 r2 r 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 185

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ZEARN MATH STUDENT EDITION G7M3 LESSON 9 Name Date GRADE 7 MISSION 3 LESSON 9 Exit Ticket Here is a picture that shows one side of a child s wooden block with a semicircle cut out at the bottom 9 cm 4 5 cm 5 cm Find the area of the side 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 187

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ZEARN MATH STUDENT EDITION G7M3 LESSON 10 Lesson 10 Distinguishing Circumference and Area Let s contrast circumference and area Warm Up 1 About how many cheese puffs can fit on the plate in a single layer Be prepared to explain your reasoning Concept Exploration ACTIVITY 1 2 Your teacher will give you cards with questions about circles Use the cards to complete the questions 1 Sort the cards into two groups based on whether you would use the circumference or the area of the circle to answer the question Pause here so your teacher can review your work 2 Your teacher will assign you a card to examine more closely What additional information would you need in order to answer the question on your card 3 Estimate measurements for the circle on your card 4 Use your estimates to calculate the answer to the question 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M3 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 190 Here are two students answers for each question Do you agree with either of them Explain or show your reasoning 1 How many feet are traveled by a person riding once around the merry go round Clare says The radius of the merry go round is about 4 feet so the distance around the edge is about 8 feet Andre says The diameter of the merry go round is about 4 feet so the distance around the edge is about 4 feet 2 How much room is there to spread frosting on the cookie Clare says The radius of the cookie is about 3 centimeters so the space for frosting is about 6 cm2 Andre says The diameter of the cookie is about 3 inches so the space for frosting is about 2 25 in2 3 How far does the unicycle move when the wheel makes 5 full rotations Clare says The diameter of the unicycle wheel is about 0 5 meters In 5 complete rotations it will go about 52 m2 Andre says I agree with Clare s estimate of the diameter but that means the unicycle will go about 54 m 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G7M3 LESSON 10 Lesson Summary Sometimes we need to find the circumference of a circle and sometimes we need to find the area Here are some examples of quantities related to the circumference of a circle The length of a circular path The distance a wheel will travel after one complete rotation The length of a piece of rope coiled in a circle Here are some examples of quantities related to the area of a circle The amount of land that is cultivated on a circular field The amount of frosting needed to cover the top of a round cake The number of tiles needed to cover a round table In both cases the radius or diameter of the circle is all that is needed to make the calculation The circumference of a circle with radius r is 2 r while its area is r2 The circumference is measured in linear units such as cm in km while the area is measured in square units such as cm2 in2 km2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 191

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ZEARN MATH STUDENT EDITION Name G7M3 LESSON 10 Date GRADE 7 MISSION 3 LESSON 10 Exit Ticket A circular lawn has a row of bricks around the edge The diameter of the lawn is about 40 feet 1 Which is the best estimate for the amount of grass in the lawn a 125 feet b 125 square feet c 1 250 feet d 1 250 square feet 2 Which is the best estimate for the total length of the bricks a 125 feet b 125 square feet c 1 250 feet d 1 250 square feet 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G7M3 LESSON 11 Lesson 11 Stained Glass Windows Let s use circumference and area to design stained glass windows Warm Up 1 The students in art class are designing a stained glass window to hang in the school entryway The window will be 3 feet tall and 4 feet wide Here is their design They have raised 100 for the project The colored glass costs 5 per square foot and the clear glass costs 2 per square foot The material they need to join the pieces of glass together costs 10 cents per foot and the frame around the window costs 4 per foot Do they have enough money to cover the cost of making the window 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7M3 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 A local community member sees the school s stained glass window and really likes the design They ask the students to create a larger copy of the window using a scale factor of 3 Would 450 be enough to buy the materials for the larger window Explain or show your reasoning ACTIVITY 2 3 196 Draw a stained glass window design that could be made for less than 450 Show your thinking Organize your work so it can be followed by 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

ZEARN MATH STUDENT EDITION G7V1 Terminology Area of a circle C E 5c m The area of a circle whose radius is r units is r2 square units A circle has radius 3 inches Its area is 32 9 square inches which is approximately 28 3 square inches G A B F D Circle A circle is made out of all the points that are the same distance from a given point For example every point on this circle is 5 cm away from point A which is the center of the circle Circumference The circumference of a circle is the distance around the circle If you imagine the circle as a piece of string it is the length of the string If the circle has radius r then the circumference is 2 r The circumference of a circle of radius 3 is 2 3 6 which is about 18 85 Constant of proportionality In a proportional relationship the values for one quantity are each multiplied by the same number to get the values for the other quantity This number is called the constant of proportionality In this example the constant of proportionality is 3 because 2 3 6 3 3 9 and 5 3 15 This means that there are 3 apples for every 1 orange in the fruit salad Number of oranges Number of apples 2 6 3 9 5 15 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 For example point B in the first triangle corresponds to point E in the second triangle Segment AC corresponds to segment DF 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7V1 ZEARN MATH STUDENT EDITION Diameter A diameter is a line segment that goes from one edge of a circle to the other and passes through the center A diameter can go in any direction Every diameter of the circle is the same length We also use the word diameter to mean the length of this segment O d For example d is the diameter of this circle with center O Equivalent ratios Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio For example 8 6 is equivalent to 4 3 because 8 12 4 and 6 12 3 A recipe for lemonade says to use 8 cups of water and 6 lemons If we use 4 cups of water and 3 lemons it will make half as much lemonade Both recipes taste the same because 8 6 and 4 3 are equivalent ratios Cups of Water Number of Lemons 8 6 4 3 Origin The origin is the point 0 0 in the coordinate plane This is where the horizontal axis and the vertical axis cross y 2 1 2 1 0 0 1 2 x 1 2 Pi There is a proportional relationship between the diameter and circumference of any circle The constant of proportionality is pi The symbol for pi is We can represent this relationship with the equation C d where C represents the circumference and d represents the diameter Some approximations for are 22 7 3 14 and 3 14159 C 12 10 8 6 4 1 2 1 2 3 4 5 6 d Proportional relationship In a proportional relationship the values for one quantity are each multiplied by the same number to get the values for the other quantity 198 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 For example in this table every value of p is equal to 4 times the value of s on the same row We can write this relationship as p 4s This equation shows that p is proportional to s G7V1 s p 2 8 3 12 5 20 10 40 Radius A radius is a line segment that goes from the center to the edge of a circle A radius can go in any direction Every radius of the circle is the same length We also use the word radius to mean the length of this segment r O For example r is the radius of this circle with center O Scale A scale tells how the measurements in a scale drawing represent the actual measurements of the object For example the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room This means that 2 inches would represent 16 feet and 12 inch would represent 4 feet Scale drawing A scale drawing represents an actual place or object All the measurements in the drawing correspond to the measurements of the actual object by the same scale For example this map is a scale drawing The scale shows that 1 cm on the map represents 30 miles on land Map of Texas and Oklahoma by United States Census Bureau via American Fact Finder Public Domain Scale factor To create a scaled copy we multiply all the lengths in the original figure by the same number This number is called the scale factor 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G7V1 ZEARN MATH STUDENT EDITION In this example the scale factor is 1 5 because 4 1 5 6 5 1 5 7 5 and 6 1 5 9 Scaled copy A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number For example triangle DEF is a scaled copy of triangle ABC Each side length on triangle ABC was multiplied by 1 5 to get the corresponding side length on triangle DEF 200 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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zearn org NAME Grade 7 Student Edition Vol 1 Mission 1 Scale Drawings Mission 2 Introducing Proportional Relationships Vol 2 Mission 4 Proportional Relationships and Percentages Mission 5 Rational Number Arithmetic Mission 6 Expressions Equations and Inequalities Student Edition Mission 3 Measuring Circles Vol 3 Mission 7 Angles Triangles and Prisms Mission 8 Probability and Sampling Mission 9 Putting It All Together G7 Vol 1 Zearnmath_SE_Grade7_Vol1 indd 1 Grade 7 Volume 1 MISSIONS 1 2 3 4 5 6 7 8 9 12 15 22 1 11 PM