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STUDENT EDITION Grade 6 VOLUME 2 Mission 4 Dividing Fractions Mission 5 Arithmetic in Base Ten Mission 6 Expressions and Equations NAME

2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum used under the CC BY 4 0 license Download the original for free at openupresources org Zearn is a registered trademark Printed in the U S A ISBN 979 8 88868 097 1

Table of Contents Mission 4 Lesson 1 Size of Divisor and Size of Quotient 3 Lesson 2 Meanings of Division 9 Lesson 3 Interpreting Division Situations 15 Lesson 4 How Many Groups Part 1 21 Lesson 5 How Many Groups Part 2 27 Lesson 6 Using Diagrams to Find the Number of Groups 33 Lesson 7 What Fraction of a Group 41 Lesson 8 How Much in Each Group Part 1 49 Lesson 9 How Much in Each Group Part 2 55 Lesson 10 Dividing by Unit and Non Unit Fractions 61 Lesson 11 Using an Algorithm to Divide Fractions 71 Lesson 12 Fractional Lengths 77 Lesson 13 Rectangles with Fractional Side Lengths 83 Lesson 14 Fractional Lengths in Triangles and Prisms 89 Lesson 15 Volume of Prisms 95 Lesson 16 Solving Problems Involving Fractions 101 Lesson 17 Fitting Boxes into Boxes 107 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license iii

Mission 5 Lesson 3 Using Decimals in a Shopping Context 113 Lesson 2 Using Diagrams to Represent Addition and Subtraction 119 Lesson 3 Adding and Subtracting Decimals with Few Non Zero Digits 125 Lesson 4 Adding and Subtracting Decimals with Many Non Zero Digits 131 Lesson 5 Decimal Points in Products 137 Lesson 6 Methods for Multiplying Decimals 143 Lesson 7 Using Diagrams to Represent Multiplication 147 Lesson 8 Calculating Products of Decimals 153 Lesson 9 Using the Partial Quotients Method 159 Lesson 10 Using Long Division 165 Lesson 11 Dividing Numbers that Result in Decimals 171 Lesson 12 Dividing Decimals by Whole Numbers 177 Lesson 13 Dividing Decimals by Decimals 185 Lesson 14 Using Operations on Decimals to Solve Problems 191 Lesson 15 Making and Measuring Boxes 197 Mission 6 iv Lesson 1 Tape Diagrams and Equations 203 Lesson 2 Truth and Equations 209 Lesson 3 Stay in Balance 215 Lesson 4 Practice Solving Equations and Representing Situations with Equations 221 Lesson 5 A New Way to Interpret a over b 227 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

Lesson 6 Write Expressions Where LettersStand for Numbers 233 Lesson 7 Revisit Percentages 239 Lesson 8 Equal and Equivalent 245 Lesson 9 The Distributive Property Part 1 253 Lesson 10 The Distributive Property Part 2 259 Lesson 11 The Distributive Property Part 3 265 Lesson 12 Meaning of Exponents 269 Lesson 13 Expressions with Exponents 275 Lesson 14 Evaluating Expressions with Exponents 281 Lesson 15 Equivalent Exponential Expressions 287 Lesson 16 Two Related Quantities Part 1 293 Lesson 17 Two Related Quantities Part 2 299 Lesson 18 More Relationships 305 Terminology 313 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license v

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Grade 6 Mission 4 Dividing Fractions

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ZEARN MATH STUDENT EDITION G6M4 LESSON 1 Lesson 1 Size of Divisor and Size of Quotient Let s explore quotients of different sizes Warm Up 1 Find the value of each expression mentally 5 000 5 5 000 2 500 5 000 10 000 5 000 500 000 Concept Exploration ACTIVITY 1 2 Here are several types of objects For each type of object estimate how many are in a stack that is 5 feet high Be prepared to explain your reasoning a Cardboard boxes b Notebooks c Bricks d Coins 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 1 3 ZEARN MATH STUDENT EDITION A stack of books is 72 inches tall Each book is 2 inches thick Which expression tells us how many books are in the stack Be prepared to explain your reasoning a 72 2 b 72 2 c 2 72 d 72 2 1 43 Another stack of books is 43 inches tall Each book is 2 inch thick Write an expression that represents the number of books in the stack ACTIVITY 2 53 63 Your teacher will give your group two sets of division expressions Without computing estimate their values and arrange each set of expressions in order from largest to smallest Be prepared to explain your reasoning Record the expressions in each set in order from largest to smallest Set 1 Set 2 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 73 30 1 2 G6M4 LESSON 1 Without computing estimate each quotient and arrange them in three groups close to 0 close to 1 and much larger than 1 Be prepared to explain your reasoning 9 10 18 9 15 000 1 500 000 30 0 45 9 10 000 18 0 18 15 000 14 500 Close to 0 Close to 1 Much larger than 1 Lesson Summary Here is a division expression 60 4 In this division we call 60 the dividend and 4 the divisor The result of the division is the quotient In this example the quotient is 15 because 60 4 15 We don t always have to make calculations to have a sense of what a quotient will be We can reason about it by looking at the size of the dividend and the divisor Let s look at some examples In 100 11 and in 18 2 9 the dividend is larger than the divisor 100 11 is very close to 99 11 which is 9 The quotient 18 2 9 is close to 18 3 or 6 In general when a larger number is divided by a smaller number the quotient is greater than 1 In 99 101 and in 7 5 7 4 the dividend and divisor are very close to each other 99 101 99 is very close to 99 100 which is 100 or 0 99 The quotient 7 5 7 4 is close to 7 5 7 5 which is 1 In general when we divide two numbers that are nearly equal to each other the quotient is close to 1 In 10 101 and in 50 198 the dividend is smaller than the divisor 10 101 is very close to 10 1 10 100 which is 100 or 0 1 The division 50 198 is close to 50 200 which is 4 or 0 25 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 5

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 1 Date GRADE 6 MISSION 4 LESSON 1 Exit Ticket Without computing decide whether the value of each expression is much smaller than 1 close to 1 or much larger than 1 1 1 000 001 99 2 3 7 4 2 3 1 835 4 100 100 5 0 006 6 000 6 50 50 1 1 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 7

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ZEARN MATH STUDENT EDITION G6M4 LESSON 2 Lesson 2 Meanings of Division Let s explore ways to think about division Warm Up 1 What are some ways to think about this expression Describe at least two meanings you think it could have 20 4 Concept Exploration ACTIVITY 1 2 A baker has 12 pounds of almonds She puts them in bags so that each bag has the same weight Clare and Tyler drew diagrams and wrote equations to show how they were thinking about 12 6 12 12 6 6 6 12 Clare s diagram and equation 2 2 6 2 2 2 2 12 Tyler s diagram and equation How do you think Clare and Tyler thought about 12 6 Explain what each diagram and each part of each equation especially the missing number might mean in the context of the bags of almonds 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 2 3 ZEARN MATH STUDENT EDITION Explain what each division expression could mean in the context of the bags of almonds Then draw a diagram and write a multiplication equation to show how you are thinking about the expression a 12 4 b 12 2 c 12 1 2 Lesson Summary Suppose 24 bagels are being distributed into boxes The expression 24 3 could be understood in two ways 24 bagels are distributed equally into 3 boxes as represented by this diagram 24 8 8 8 24 bagels are distributed into boxes 3 bagels in each box as represented by this diagram 24 3 3 3 3 3 3 3 3 In both interpretations the quotient is the same 24 3 8 but it has different meanings in each case In the first case the 8 represents the number of bagels in each of the three boxes In the second it represents the number of boxes that were formed with 3 bagels in each box These two ways of seeing division are related to how 3 8 and 24 are related in multiplication Both 3 8 and 8 3 equal 24 10 3 8 24 can be read as 3 groups of 8 make 24 8 3 24 can be read as 8 groups of 3 make 24 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 2 If 3 and 24 are the only numbers given the multiplication equations would be 3 24 3 24 In both cases the division 24 3 can be used to find the value of the But now we see that it can be interpreted in more than one way because the can refer to the size of a group as in 3 groups of what number make 24 or to the number of groups as in How many groups of 3 make 24 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 11

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ZEARN MATH STUDENT EDITION G6M4 LESSON 2 Name Date GRADE 6 MISSION 4 LESSON 2 Exit Ticket 1 During a field trip 60 students are put into equal sized groups a Describe two ways to interpret 60 5 in this context b Find the quotient c 2 Explain what the quotient would mean in each of the two interpretations you described Consider the division expression 7 this division expression a 2 7 b 7 c 1 2 2 7 d 7 1 2 2 Select all multiplication equations that correspond to 1 2 2 1 2 1 2 2 e 2 7 1 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 13

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ZEARN MATH STUDENT EDITION G6M4 LESSON 3 Lesson 3 Interpreting Division Situations Let s explore situations that involve division Warm Up 1 How many dots are in the dot pattern Explain how you saw them Concept Exploration ACTIVITY 1 2 Draw a diagram and write a multiplication equation to represent each of the following situations Then answer the question about each situation 1 Mai had 4 jars In each jar she put 2 cups of jam are in the jars 2 Priya filled 5 jars using a total of 7 3 Han had some jars He put did he fill 3 4 1 2 1 4 cups of homemade blueberry jam Altogether how many cups of strawberry jam How many cups of jam are in each jar cup of grape jam in each jar using a total of 6 3 4 cups How many jars 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 3 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 1 Answer these questions about Kiran making granola To make 1 batch of granola Kiran needs 26 ounces of oats The only measuring tool he has is a 4 ounce scoop How many scoops will it take to measure 26 ounces of oats a Will the answer be more than 1 or less than 1 b Write a multiplication equation and a division equation that represent this situation Use to represent the unknown quantity c 2 Find the unknown quantity If you get stuck draw a diagram The recipe calls for 14 ounces of mixed nuts To get that amount Kiran uses 4 bags of mixed nuts a Write a mathematical question that might be asked about this situation b What might the equation 14 4 represent in Kiran s situation c 16 Find the quotient Show your reasoning If you get stuck draw a diagram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M4 LESSON 3 Lesson Summary If a situation involves equal sized groups it is helpful to make sense of it in terms of the number of groups the size of each group and the total amount Here are three examples to help us better understand such situations Suppose we have 3 bottles with 6 12 ounces of water in each and the total amount of water is not given Here we have 3 groups 6 12 ounces in each group and an unknown total as shown in this diagram 6 12 6 12 6 12 We can express this situation as a multiplication problem The unknown is the product so we can simply multiply the 2 known numbers to find it 3 6 Next suppose we have 20 ounces of water to fill 6 equal sized bottles and the amount in each bottle is not given Here we have 6 groups an unknown amount in each and a total of 20 We can represent it like this 20 1 2 This situation can also be expressed using multiplication but the unknown is a factor rather than the product 6 20 To find the unknown we cannot simply multiply but we can think of it as a division problem 20 6 40 Now suppose we have 40 ounces of water to pour into bottles 12 ounces in each bottle but the number of bottles is not given Here we have an unknown 12 12 number of groups 12 in each group and a total of 40 Again we can think of this in terms of multiplication with a different factor being the unknown 12 40 Likewise we can use division to find the unknown 40 12 Whenever we have a multiplication situation one factor tells us how many groups there are and the other factor tells us how much is in each group Sometimes we want to find the total Sometimes we want to find how many groups there are Sometimes we want to find how much is in each group Anytime we want to find out how many groups there are or how much is in each group we can represent the situation using division 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 17

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 3 Date GRADE 6 MISSION 4 LESSON 3 Exit Ticket 1 Here are three problems Select all problems that can be solved using division a Jada cut 4 pieces of ribbon that were equal in length She used a total of 5 feet of ribbon How long in feet was each piece of ribbon she cut b A chef bought 3 bags of beans Each bag contains 1 kilograms of beans did she buy c 2 2 5 kilograms of beans How many A printer takes 2 12 seconds to print a flyer It took 75 seconds to print a batch of flyers without stopping How many flyers were in the batch Andre poured 27 ounces of rice into 6 containers If all containers have the same amount of rice how many ounces are in each container a Write an equation to represent the situation Use a to represent the unknown quantity b Find the unknown quantity 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 19

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ZEARN MATH STUDENT EDITION G6M4 LESSON 4 Lesson 4 How Many Groups Part 1 Let s play with blocks and diagrams to think about division with fractions Warm Up 1 Write a multiplication equation and a division equation for each statement or diagram 1 Eight 5 bills are worth 40 2 There are 9 thirds in 3 ones 3 1 1 5 1 5 1 5 1 5 1 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 21

G6M4 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Your teacher will give you pattern blocks as shown here Use them to answer the following questions If a hexagon represents 1 whole what fraction do each of the following shapes represent Be prepared to show or explain your reasoning a 1 triangle b 1 rhombus c 1 trapezoid d 4 triangles e 3 rhombuses f 2 hexagons g 1 hexagon and 1 trapezoid 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 3 G6M4 LESSON 4 Here are Elena s diagrams for 2 12 and 6 13 2 2 12 1 6 13 2 1 Do you think these diagrams represent the equations Explain or show your reasoning 2 Use pattern blocks to represent each multiplication equation Recall that a hexagon represents 1 whole 3 a 3 1 6 1 2 b 2 3 2 3 Answer the following questions If you get stuck use pattern blocks a How many 1 2 s are in 4 b How many 2 3 s are in 2 c 1 6 s are in 1 12 How many 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary Some problems that involve equal sized groups also involve fractions Here is an example How many 1 6 s are in 2 We can express this question with multiplication and division equations 1 6 2 2 1 6 Pattern block diagrams can help us make sense of such problems Here is a set of pattern blocks If the hexagon represents 1 whole then a triangle must represent 16 because 6 triangles make 1 hexagon We can use the triangle to represent the 16 in the problem Twelve triangles make 2 hexagons which means there are 12 groups of 1 6 in 2 If we write the 12 in the place of the in the original equations we have 12 16 2 2 24 1 6 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 G6M4 LESSON 4 Name Date GRADE 6 MISSION 4 LESSON 4 Exit Ticket 1 The hexagon represents 1 whole Draw a pattern block diagram that represents the equation 4 2 1 3 1 13 Answer the following questions If you get stuck use pattern blocks a How many 1 2 s are in 3 12 b How many 1 3 s are in 2 23 c 1 6 s are in How many 2 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 25

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ZEARN MATH STUDENT EDITION G6M4 LESSON 5 Lesson 5 How Many Groups Part 2 Let s use blocks and diagrams to understand more about division with fractions Warm Up 1 Write a fraction or whole number as an answer for each question If you get stuck use the fraction strips Be prepared to share your strategy 1 How many 1 2 s are in 2 2 How many 1 5 s are in 3 3 How many 1 8 s are in 1 14 4 1 2 6 5 2 2 9 6 2 4 10 1 1 1 2 1 2 1 3 1 4 1 5 1 6 1 1 8 8 1 1 9 9 1 3 1 4 1 6 1 5 1 3 1 4 1 5 1 6 1 1 8 8 1 1 1 9 9 9 1 5 1 1 6 6 1 1 8 8 1 1 9 9 1 2 1 2 1 4 1 5 1 6 1 1 8 8 1 1 9 9 1 3 1 4 1 5 1 6 1 1 8 8 1 1 9 9 1 3 1 4 1 6 1 5 1 3 1 4 1 5 1 6 1 1 8 8 1 1 1 9 9 9 1 5 1 1 6 6 1 1 8 8 1 1 9 9 1 4 1 5 1 6 1 1 8 8 1 1 9 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 27

G6M4 LESSON 5 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Your teacher will give you pattern blocks as shown here Use them to answer the following questions If the trapezoid represents 1 whole what do each of the following shapes represent Be prepared to show or explain your reasoning a 1 triangle b 1 rhombus c 2 28 1 hexagon Use pattern blocks to represent each multiplication equation Use the trapezoid to represent 1 whole a 3 1 3 1 b 3 2 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 Diego and Jada were asked How many rhombuses are in a trapezoid Diego says 1 13 If I put 1 rhombus on a trapezoid the leftover shape is a triangle which is of the trapezoid Jada says I think it s 1 12 Since we want to find out how many rhombuses we should compare the leftover triangle to a rhombus A triangle is 12 of a rhombus Is the answer 1 4 G6M4 LESSON 5 1 3 or 1 1 2 1 3 Show or explain your reasoning Select all equations that can be used to answer the question How many rhombuses are in a trapezoid a 2 3 1 b c 1 2 3 2 3 1 d 1 2 3 e 2 3 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 29

G6M4 LESSON 5 ZEARN MATH STUDENT EDITION Lesson Summary Suppose one batch of cookies requires flour 2 3 cup flour How many batches can be made with 4 cups of We can think of the question as being How many and division equations 23 4 and 4 23 2 3 s are in 4 and represent it using multiplication Let s use pattern blocks to visualize the situation and say that a hexagon is 1 whole 2 1 3 4 5 6 Since 3 rhombuses make a hexagon 1 rhombus represents 13 and 2 rhombuses represent see that 6 pairs of rhombuses make 4 hexagons so there are 6 groups of 23 in 4 2 3 We can Other kinds of diagrams can also help us reason about equal sized groups involving fractions This example shows how we might reason about the same question from above How many 23 cups are in 4 cups 1 3 1 3 cup cup We can see each cup partitioned into thirds and that there are 6 groups of 23 cup in 4 cups In both diagrams we see that unknown value or the in the equations is 6 So we can now write 6 23 4 and 4 23 6 30 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 5 Name Date GRADE 6 MISSION 4 LESSON 5 Exit Ticket 2 A grocery store sells tangerines in 5 kg bags A customer bought 4 kg of tangerines for a school party How many bags did he buy 1 Select all equations that represent the situation a 4 2 5 b 2 5 4 c 2 2 5 4 d 4 2 5 e 2 5 4 Draw a diagram to represent the situation Answer 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 31

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ZEARN MATH STUDENT EDITION G6M4 LESSON 6 Lesson 6 Using Diagrams to Find the Number of Groups Let s draw tape diagrams to think about division with fractions Warm Up 1 1 Complete the tape diagrams to represent the following questions We can think of the division expression 10 2 12 as the answer to the question How many groups of 2 12 are in 10 Complete the tape diagram to represent the question Then answer the question 10 2 Complete the tape diagram to represent the question How many groups of 2 are in 7 Then answer the question 7 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 33

G6M4 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 2 3 To make sense of the question How many and drew a tape diagram s are in 1 Andre wrote equations 1 2 3 1 1 2 3 1 3 1 3 1 group of 2 3 1 In an earlier task we used pattern blocks to help us solve the equation 1 Andre s tape diagram can also help us solve the equation 2 Write a multiplication equation and a division equation for each of the following questions Draw a tape diagram to find the solution Use the grid to help you draw if needed a How many 34 2 3 3 4 Explain how s are in 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 G6M4 LESSON 6 b How many 2 3 s are in 3 c 3 2 s are in 5 How many ACTIVITY 2 3 For each question draw a diagram to show the relationship of the quantities and to help you answer the question Then write a multiplication equation or a division equation for the situation described in the question Be prepared to share your reasoning a How many 3 8 inch thick books make a stack that is 6 inches tall b How many groups of 1 2 pound are in 2 3 4 pounds 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 35

G6M4 LESSON 6 43 36 ZEARN MATH STUDENT EDITION Write a question that can be represented by the division equation 5 1 12 Then answer the question Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M4 LESSON 6 Lesson Summary A baker used 2 kilograms of flour to make several batches of a pastry recipe The recipe called for kilogram of flour per batch How many batches did she make 2 5 We can think of the question as How many groups of that question with the equations 2 5 2 2 2 5 2 5 kilogram make 2 kilograms and represent To help us make sense of the question we can draw a tape diagram This diagram shows 2 whole kilograms with each kilogram partitioned into fifths 2 kg 1 5 1 5 1 batch batches We can see there are 5 groups of 25 in 2 Multiplying 5 and 10 5 25 10 5 and 5 2 so the answer is correct 2 5 allows us to check this answer Notice the number of groups that result from 2 25 is a whole number Sometimes the number of groups we find from dividing may not be a whole number Here is an example Suppose one serving of rice is 3 4 cup How many servings are there in 3 1 2 cups 3 12 cups 1 serving servings 3 1 2 3 4 3 3 4 1 2 Looking at the diagram we can see there are 4 full groups of 34 plus 2 fourths If 3 fourths make a whole group then 2 fourths make 23 of a group So the number of servings the in each equation is 4 23 We can check this by multiplying 4 23 and 34 4 2 3 3 4 14 3 3 4 and 14 3 3 4 1 14 4 which is indeed equivalent to 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 37

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 6 Date GRADE 6 MISSION 4 LESSON 6 Exit Ticket How many 3 4 s are in 2 1 Write a multiplication equation and a division equation that can be used to answer the question 2 Draw a tape diagram and answer the question Use the grid to help you draw if needed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 39

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ZEARN MATH STUDENT EDITION G6M4 LESSON 7 Lesson 7 What Fraction of a Group Let s think about dividing things into groups when we can t even make one whole group Warm Up 1 Estimate the following quantities Then write a multiplication expression for each question a What is 1 3 of 7 b What is 4 5 of 9 23 c What is 2 47 of 10 19 Concept Exploration ACTIVITY 1 2 Here is a diagram that shows four ropes of different lengths A B C D 1 Compare the lengths of rope B C and D to the length of rope A and complete each statement Then use the measurements shown on the grid to write a multiplication equation and a division equation for each statement a Rope B is as long as rope A times b Rope C is as long as rope A times c Rope D is as long as rope A times 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 7 2 ZEARN MATH STUDENT EDITION Each equation can be used to answer a question about rope C and D What could each question be a 3 9 and 9 3 b 9 3 and 3 9 ACTIVITY 2 3 One batch of an ice cream recipe uses 9 cups of milk A chef makes different amounts of ice cream on different days Here are the amounts of milk she used 1 Monday 12 cups Tuesday 22 Thursday 6 cups Friday 7 12 cups 1 2 cups How many batches of ice cream did she make on each of the following days Write a division equation and draw a tape diagram for the question about each day Then answer the question a Monday b Tuesday 42 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 2 G6M4 LESSON 7 What fraction of a batch of ice cream did she make on each of the following days Write a division equation and draw a tape diagram for the question about each day Then answer the question a Thursday b Friday 43 Write a division equation and draw a tape diagram for each question Then answer the question a What fraction of 9 is 3 b What fraction of 5 is 1 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 43

G6M4 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary It is natural to think about groups when we have more than one group but we can also have a fraction of a group To find the amount in a fraction of a group we can multiply the fraction by the amount in the whole group If a bag of rice weighs 5 kg 34 of a bag would weigh 34 5 kg 5 kg 34 5 kg 3 4 bag 1 bag 3 cups 1 2 4 or 9 4 cups group 1 group Sometimes we need to find what fraction group an amount is Suppose a full bag of flour weighs 6 kg A chef used 3 kg of flour What fraction of a full bag was used In other words what fraction of 6 kg is 3 kg This question can be represented by a multiplication equation and a division equation as well as by a diagram 6 3 3 6 6 kg 3 kg bag 1 bag 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 G6M4 LESSON 7 We can see from the diagram that 3 is 1 2 of 6 and we can check this answer by multiplying 12 6 3 In any situation where we want to know what fraction of one number is of another number we can write a division equation to help us find the answer For example What fraction of 3 is 2 14 can be expressed as 3 2 14 which can also be written as 2 1 4 3 The answer to What is 2 1 4 3 is also the answer to the original question 1 4 The diagram shows that 3 wholes contain 12 fourths and 2 9 question is 12 which is equivalent to 34 contains 9 fourths so the answer to this We can use diagrams to help us solve other division problems that require finding a fraction of a group 9 3 For example here is a diagram to help us answer the question What fraction of 4 is 2 which can be 3 9 written as 2 4 1 2 4 or 3 2 or 9 4 cups 6 4 group 1 group We can see that the quotient is To check this let s multiply 2 3 6 9 which is equivalent to 9 4 2 3 18 18 12 and 12 is indeed equal to 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 45

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 7 Date GRADE 6 MISSION 4 LESSON 7 Exit Ticket There is 1 3 gallon of water in a 3 gallon container What fraction of the container is filled 1 Write a multiplication equation and a division equation to represent the situation 2 Draw a tape diagram to represent the situation Then answer 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 47

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ZEARN MATH STUDENT EDITION G6M4 LESSON 8 Lesson 8 How Much in Each Group Part 1 Let s look at division problems that help us find the size of one group Warm Up 1 Think of a situation with a question that can be represented by 12 1 Write a description of that situation and the question 2 Trade descriptions with your partner and answer your partner s question 2 3 Concept Exploration ACTIVITY 1 2 For each question write a multiplication equation and a division equation draw a diagram and answer the question We can write equations and draw a diagram to represent this situation They help us see that each batch requires 2 cups of flour 1 To make 4 batches of cupcakes it takes 6 cups of flour How many cups of flour are needed for 1 batch 2 To make 1 2 batch of rolls it takes 5 4 cups of flour How many cups of flour are needed for 1 batch 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 8 3 Two cups of flour make ZEARN MATH STUDENT EDITION 2 3 batch of bread How many cups of flour make 1 batch ACTIVITY 2 Here are three tape diagrams and three descriptions of situations that include questions Match a diagram to each situation then use the diagram to help you answer the question Next write multiplication and division equations to represent each situation 3 Diagram 1 Diagram 2 15 cups 15 cups Diagram 3 15 cups 1 container 1 1 container 1 container Tyler poured 15 cups of water into 2 equal sized bottles and filled each bottle How much water was in each bottle Diagram Answer Multiplication equation Division equation 2 Kiran poured 15 cups of water into equal sized pitchers and filled pitchers How much water was in the full pitcher Diagram Answer Multiplication equation Division equation 50 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 8 It takes 15 cups of water to fill 1 3 pail How much water is needed to fill 1 pail Diagram Answer Multiplication equation Division equation Lesson Summary Sometimes we know the amount for multiple groups but we don t know how much is in one group We can use division to find out For example If 5 people share 8 does each person get 1 2 pounds of cherries equally how many pounds of cherries 8 12 pounds 1 person 5 people We can represent this situation as a multiplication and a division 5 8 8 1 2 1 2 5 8 12 5 can be written as 17 5 Dividing by 5 is equivalent to multiplying by 2 7 pounds This means each person gets 1 10 1 5 and 17 2 1 5 17 10 Other times we know the amount for a fraction of a group but we don t know the size of one whole group We can also use division to find out cups 5 cups 2 3 pitcher 1 pitcher For example Jada poured 5 cups of iced tea in a pitcher and filled iced tea fill the entire pitcher 2 3 of the pitcher How many cups of 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 51

G6M4 LESSON 8 ZEARN MATH STUDENT EDITION We can represent this situation as a multiplication and a division 2 3 5 5 2 3 The diagram can help us reason about the answer If 23 of a pitcher is 5 cups then 13 of a pitcher is half of 5 which is 25 Because there are 3 thirds in 1 whole there would be 3 52 or 15 cups in one whole 2 2 15 30 30 pitcher We can check our answer by multiplying 3 2 6 and 6 5 Notice that in the first example the number of groups is greater than 1 5 people and in the second the number of groups is less than 1 23 of a pitcher but the division and multiplication equations for both have the same structures 52 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G6M4 LESSON 8 Date GRADE 6 MISSION 4 LESSON 8 Exit Ticket Students in a sixth grade class are raising money for an end of year camping trip So far they 2 have raised 240 This is 5 of the cost of the trip How much does the trip cost Write multiplication and division equations and draw a diagram to represent the situation Then answer the question and 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 53

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ZEARN MATH STUDENT EDITION G6M4 LESSON 9 Lesson 9 How Much in Each Group Part 2 Let s practice dividing fractions in different situations Warm Up Decide whether each of the following is greater than 1 or less than 1 1 1 1 2 2 1 3 2 3 4 2 3 4 7 8 1 4 7 8 2 3 5 Concept Exploration ACTIVITY 1 2 1 2 Write a multiplication equation and a division equation and draw a diagram to represent each situation and question Then find the answer Explain your reasoning Jada bought 3 4 9 1 2 yards of fabric for 21 How much did each yard cost kilogram of baking soda costs 2 How much does 1 kilogram of baking soda cost 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 9 3 4 ZEARN MATH STUDENT EDITION Diego can fill 1 5 4 1 5 bottles with 3 liters of water How many liters of water fill 1 bottle gallons of water fill 5 6 of a bucket How many gallons of water fill the entire bucket ACTIVITY 2 3 56 Think of a situation that involves a question that can be represented by 1 3 1 4 Write a description of that situation and 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

ZEARN MATH STUDENT EDITION G6M4 LESSON 9 Lesson Summary Sometimes we have to think carefully about how to solve a problem that involves multiplication and division Diagrams and equations can help us 3 4 Let s take this example of a pound of rice fills 2 5 of a container There are two whole amounts to keep track of 1 whole pound and 1 whole container The equations we write and the diagram we draw depend on what question we are trying to answer Here are two questions that could be asked How many pounds fill 1 container What fraction of a container does 1 pound fill We can represent and answer the first question how many pounds fill a whole container with pounds 3 4 2 5 3 4 2 5 pound 3 4 2 5 container 1 container If 25 of a container is filled with 34 pound then 15 of a container is filled with half of One whole container then has 5 38 or 15 8 pounds 3 4 or 3 8 pound We can represent and answer the second question what fraction of the container 1 pound fills with container 2 5 3 4 2 5 3 4 container 2 5 3 4 pound 1 pound If 34 pound fills 25 of a container then 8 2 then fills 4 15 or 15 of a container 1 4 pound fills a third of 2 5 2 or 15 of a container One whole pound 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 57

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ZEARN MATH STUDENT EDITION G6M4 LESSON 9 Name Date GRADE 6 MISSION 4 LESSON 9 Exit Ticket Noah fills a soap dispenser from a big bottle that contains 2 amount of soap will fill 3 1 2 1 3 liters of liquid soap That dispensers How many liters of soap fit into one dispenser Use the diagram below to answer the question Label all relevant parts of the diagram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 59

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ZEARN MATH STUDENT EDITION G6M4 LESSON 10 Lesson 10 Dividing by Unit and Non Unit Fractions Let s look for patterns when we divide by a fraction Warm Up 1 1 Work with a partner on the following problems One person should solve the problems labeled Partner A and the other should solve those labeled Partner B Write an equation for each question Consider drawing a diagram Partner A a How many 3s are in 12 Division equation b How many 4s are in 12 Division equation c How many 6s are in 12 Division equation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 61

G6M4 LESSON 10 2 ZEARN MATH STUDENT EDITION Partner B a What is 12 groups of 13 Multiplication equation b What is 12 groups of 14 Multiplication equation c 62 1 What is 12 groups of 6 Multiplication equation 3 What do you notice in the diagrams and equations Discuss with your partner 4 Complete this sentence based on your observations Dividing by a whole number a produces the same result as multiplying 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 G6M4 LESSON 10 Concept Exploration ACTIVITY 1 1 2 1 To find the value of 6 2 Elena thought How many 2 s are in 6 and drew a tape diagram It shows 6 ones with each one partitioned into 2 equal pieces 6 1 2 1 group group 1 For each division expression complete the diagram using the same interpretation of division as Elena s Then write the value of the expression Think about how to find that value without counting the pieces in the diagram a 6 1 3 6 Value of the expression b 6 1 4 6 Value of the expression c 6 1 6 6 Value of the expression 2 Analyze the expressions and your answers Look for a pattern How did you find how many 1 1 1 3 s 4 s or 6 s were in 6 without counting Explain your reasoning 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 63

G6M4 LESSON 10 3 ZEARN MATH STUDENT EDITION Use your observations from previous questions to find the values of the following expressions If you get stuck you can draw diagrams a 6 1 8 1 b 6 10 4 1 6 25 c d 6 1 b c a 1 5 d a 1 b Find the value of each expression a 8 1 4 b 12 1 5 ACTIVITY 2 TASK 1 3 To find the value of 6 she did for 6 1 3 2 3 Elena began by drawing her diagram in the same way a Use her diagram to find out how many 2 3 s are in 6 Adjust and label the diagram as needed 6 1 3 b She says To find 6 23 I can just take the value of 6 2 Do you agree with her Explain why or why not 64 1 3 then either multiply it by 1 2 or divide it 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 G6M4 LESSON 10 ACTIVITY 2 TASK 2 Solve the following problems about dividing fractions 43 1 For each division expression complete the diagram using the same interpretation of division that Elena did Then write the value of the expression Think about how you could find the value of each expression without counting the equal pieces in your diagram 6 3 4 6 1 4 6 Value of the expression 4 3 6 1 3 6 Value of the expression 4 6 6 1 6 2 Value of the expression Elena studied her diagrams and noticed that she always took the same two steps to represent division by a fraction on a tape diagram She said My first step was to partition each 1 whole into as many parts as the number in the denominator So if the expression is 6 34 I would partition each 1 whole into 4 parts Now I have 4 times as many parts My second step was to put a certain number of those parts into one group and that number is the numerator of the divisor So if the fraction is 34 I would put 3 of the 14 s into one group I could then tell how many 34 s are in 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 65

G6M4 LESSON 10 ZEARN MATH STUDENT EDITION Which expression represents how many to explain your reasoning a 6 4 3 3 s Elena would have after these two steps Be prepared b 6 4 3 c 6 4 3 d 6 4 3 Use your work from the previous questions to find the values of the following expressions Draw diagrams if you are stuck a 6 66 3 4 2 7 3 b 6 10 c 6 6 25 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 10 Lesson Summary To answer the question How many 13 s are in 4 or What is 4 thirds in 1 so there are 4 3 thirds in 4 In other words dividing 4 by 1 3 1 3 we can reason that there are 3 has the same outcome as multiplying 4 by 3 4 1 3 4 1 4 3 3 1 group group In general dividing a number by a unit fraction the reciprocal of b1 How can we reason about 4 2 3 1 b is the same as multiplying the number by b which is We already know that there are 4 3 or 12 groups of 13 s in 4 To find how many 23 s are in 4 we need to put together every 2 of the 13 s into a group Doing this results in half as many groups which is 6 groups In other words 4 1 3 1 3 4 or 4 1 group group 2 4 3 2 3 1 2 4 3 2 3 In general dividing a number by ab is the same as multiplying the number by b and then dividing by a or multiplying the number by b and then by a1 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 67

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ZEARN MATH STUDENT EDITION G6M4 LESSON 10 Name Date GRADE 6 MISSION 4 LESSON 10 Exit Ticket 1 Explain or show how you could find 5 support your reasoning 1 3 by using the value of 5 3 If needed use this diagram to 5 2 Find 12 3 5 Only use a diagram if necessary 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 69

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ZEARN MATH STUDENT EDITION G6M4 LESSON 11 Lesson 11 Using an Algorithm to Divide Fractions Let s divide fractions using the rule we learned Warm Up Evaluate each expression 1 1 2 3 27 3 2 9 5 3 5 1 34 2 1 2 4 27 100 2 3 200 9 5 7 Concept Exploration ACTIVITY 1 2 1 Work with a partner One person should work on the questions labeled Partner A and the other should work on those labeled Partner B Partner A Find the value of each expression and answer the question by completing the diagram that has been started for you Show your reasoning a 3 4 18 How many 1 8 s in 3 4 3 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 71

G6M4 LESSON 11 b 9 10 35 How many ZEARN MATH STUDENT EDITION 3 5 9 s in 10 9 10 3 Partner B Elena said If you want to divide 4 by 25 you can multiply 4 by 5 then divide it by 2 or multiply it by 12 Find the value of each expression using the strategy that Elena described a 3 3 4 3 4 b 9 10 3 5 Solve the following problems about dividing fractions Complete this statement based on your observations To divide a number n by a fraction ab we can multiply n by by and then divide the product Select all equations that represent the statement you completed a n b n c n d n 72 1 8 a b a b a b a b n b a n a b n n a b 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

ZEARN MATH STUDENT EDITION G6M4 LESSON 11 ACTIVITY 2 Calculate each quotient using your preferred strategy Show your work and be prepared to explain your strategy 43 a 8 9 c 3 1 3 e 6 2 5 3 53 4 2 9 b 3 4 1 2 d 9 2 3 8 After biking 5 12 miles Jada has traveled 23 of the length of her trip How long in miles is the entire length of her trip Write an equation to represent the situation and find the answer using your preferred strategy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 11 ZEARN MATH STUDENT EDITION Lesson Summary The division a 34 is equivalent to 34 a so we can think of it as meaning 34 of what number is a and represent it with a diagram as shown The length of the entire diagram represents the unknown number 9 10 If 34 of a number is a then to find the number we can first divide a by 3 to find we multiply the result by 4 to find the number 1 4 of the number Then The steps above can be written as a 3 4 Dividing by 3 is the same as multiplying by also write the steps as a 13 4 In other words a 3 4 a 1 3 4 And a 1 3 4 a a In general dividing a number by a fraction reciprocal of the fraction 74 c d 3 4 4 3 a 1 3 so we can so we can say that 4 3 is the same as multiplying the number by d c which is the 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 11 Name Date GRADE 6 MISSION 4 LESSON 11 Exit Ticket 1 Find the value of 24 25 4 5 2 If 43 liters of water are enough to water 25 of the plants in the house how much water is necessary to water all the plants in the house Write a multiplication equation and a division equation for the situation then answer the question Show your reasoning 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 75

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ZEARN MATH STUDENT EDITION G6M4 LESSON 12 Lesson 12 Fractional Lengths Let s solve problems about fractional lengths Warm Up 1 Find the product mentally 19 14 Concept Exploration ACTIVITY 1 2 Solve the following problems 1 Jada was using square stickers with a side length of 34 inch to decorate the spine of a photo album The spine is 10 12 inches long If she laid the stickers side by side without gaps or overlaps how many stickers did she use to cover the length of the spine 2 How many 3 It takes exactly 26 paper clips laid end to end to make a length of 17 5 8 inch binder clips laid side by side make a length of 11 7 8 1 4 inches inches a Estimate the length of each paper clip b Calculate the length of each paper clip 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 77

G6M4 LESSON 12 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 1 Solve the following problems with your group A second grade student is 4 feet tall Her teacher is 5 2 3 feet tall a How many times as tall as the student is the teacher b What fraction of the teacher s height is the student s height 2 Find each quotient Show your reasoning and check your answer a 9 3 5 7 8 b 1 3 3 4 Write a division expression that can help answer each of the following questions Then answer the question If you get stuck draw a diagram 3 a A runner ran 1 45 miles on Monday and 6 10 miles on Tuesday How many times her Monday s distance was her Tuesday s distance b A cyclist planned to ride 9 12 miles but only managed to travel 3 planned trip did he travel 78 7 8 miles What fraction of his 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 12 Lesson Summary Division can help us solve comparison problems in which we find out how many times as large or as small one number is compared to another Here is an example A student is playing two songs for a music recital The first song is 1 3 34 minutes long 1 2 minutes long The second song is 1 1 minutes 2 First song Second song 3 3 minutes 4 We can ask two different comparison questions and write different multiplication and division equations to represent each question How many times as long as the first song is the second song 1 3 3 4 1 2 3 3 4 3 1 2 1 1 Let s use the algorithm we learned to calculate the quotient 3 3 4 1 15 4 15 4 1 2 3 2 2 3 30 12 What fraction of the second song is the first song 1 1 2 3 4 3 Let s calculate the quotient 1 1 2 3 3 4 3 2 15 4 3 2 4 15 12 30 5 2 This means the second song is 2 as long as the first song 1 2 3 4 1 2 times 2 5 The first song is 2 5 as long as the second song 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 12 Date GRADE 6 MISSION 4 LESSON 12 Exit Ticket A builder was building a fence In the morning he worked for 25 of an hour In the afternoon he worked for 109 of an hour How many times as long as in the morning did he work in the afternoon Write a division equation to represent this situation then answer the question Show your reasoning If you get stuck you can draw a diagram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 81

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ZEARN MATH STUDENT EDITION G6M4 LESSON 13 Lesson 13 Rectangles with Fractional Side Lengths Let s explore rectangles that have fractional measurements Warm Up 1 Use the squares to answer the questions 1 2 1 in 1 in 1 2 in 2 in in 2 in 1 What do you notice about the areas of the squares Write your observations 2 Consider the statement A square with side lengths of 13 inch has an area of Do you agree or disagree with the statement Explain your reasoning 1 3 square 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 83

G6M4 LESSON 13 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Use one piece of 1 4 inch graph paper for the following Use a ruler to draw a square with side length of 1 inch on the graph paper Inside the square draw a square with side length of 14 inch a How many squares with side length of 1 4 inch can fit in a square with side length of 1 inch b What is the area of a square with side length of 2 3 84 1 4 inch Explain how you know Use a ruler to draw a rectangle that is 3 12 inches by 2 14 inches on the graph paper Write a division expression for each question and answer the question a How many 1 4 inch segments are in a length of 3 1 2 inches b How many 1 4 inch segments are in a length of 2 1 4 inches Use your drawings to show that a rectangle that is 3 square inches 1 2 inches by 2 1 4 inches has an area of 7 7 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 13 ACTIVITY 2 3 Noah would like to cover a rectangular tray with rectangular tiles The tray has a width of 11 14 inches and an area of 50 58 square inches 1 Find the length of the tray in inches 2 9 If the tiles are 34 inch by 16 inch how many would Noah need to cover the tray completely without gaps or overlaps Explain or show your reasoning 3 Draw a diagram to show how Noah could lay the tiles Your diagram should show how many tiles would be needed to cover the length and width of the tray but does not need to show every tile 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 13 ZEARN MATH STUDENT EDITION Lesson Summary If a rectangle has side lengths a units and b units the area is a b square units For example if we have a rectangle with 12 inch side lengths its area is 12 12 or 14 square inches 1 2 1 2 in in 1 in This means that if we know the area and one side length of a rectangle we can divide to find the other side length 1 10 2 in 1 If one side length of a rectangle is 10 their relationship 89 4 in2 1 2 in and its area is 89 10 1 2 89 1 4 in2 we can write this equation to show 1 4 Then we can find the other side length in inches using division 89 86 1 4 10 1 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

ZEARN MATH STUDENT EDITION Name G6M4 LESSON 13 Date GRADE 6 MISSION 4 LESSON 13 Exit Ticket Two rectangular picture frames have the same area of 45 square inches but have different side lengths Frame A has a length of 6 34 inches and Frame B has a length of 7 12 inches 1 Without calculating predict which frame has the shorter width Explain your reasoning 2 Find the width that you predicted to be shorter 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 87

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ZEARN MATH STUDENT EDITION G6M4 LESSON 14 Lesson 14 Fractional Lengths in Triangles and Prisms Let s explore area and volume when fractions are involved Warm Up 1 Find the area of Triangle A in square centimeters Show your reasoning 4 A 4 1 2 1 2 cm cm Concept Exploration ACTIVITY 1 2 1 Find the missing measurement for each triangle The area of Triangle B is 8 square units Find the length of b Show your reasoning 8 3 B 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 89

G6M4 LESSON 14 2 ZEARN MATH STUDENT EDITION The area of Triangle C is 54 5 square units What is the length of h Show your reasoning 3 35 C h ACTIVITY 2 3 Your teacher will give you a set of cubes with an edge length of to help you answer the following questions 1 2 inch Use them 1 a Here is a drawing of a cube with an edge length of 1 inch How many cubes with an edge 1 length of 2 inch are needed to fill this cube 1 in 1 in 1 in b What is the volume in cubic inches of a cube with an edge length of your reasoning c 90 1 2 inch Explain or show Four cubes are piled in a single stack to make a prism Each cube has an edge length of inch Sketch the prism and find its volume in cubic inches 1 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

ZEARN MATH STUDENT EDITION 2 G6M4 LESSON 14 Use cubes with an edge length of shown in the table 1 2 inch to build prisms with the lengths widths and heights 1 2 a For each prism record in the table how many the volume of the prism inch cubes can be packed into the prism and Prism length in Prism width in Prism height in 1 2 1 2 1 2 1 1 1 2 2 1 1 2 2 2 1 4 2 3 2 5 4 2 5 4 2 Number of 12 cubes in prism Volume of prism cu in 1 2 b Analyze the values in the table What do you notice about the relationship between the edge lengths of each prism and its volume 3 What is the volume of a rectangular prism that is 1 reasoning 1 2 inches by 2 1 4 inches by 4 inches Show your 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 14 ZEARN MATH STUDENT EDITION Lesson Summary If a rectangular prism has edge lengths of 2 units 3 units and 5 units we can think of it as 2 layers of unit cubes with each layer having 3 5 unit cubes in it So the volume in cubic units is 2 3 5 To find the volume of a rectangular prism with fractional edge lengths we can think of it as being built of cubes that have a unit fraction for their edge length For instance if we build a prism that is 12 inch tall 32 inch wide and 4 inches long using cubes with a 12 inch edge length we would have 1 1 2 2 A width of 3 cubes because 3 12 32 A length of 8 cubes because 8 12 4 A height of 1 cube because 1 The volume of the prism would be 1 3 8 or 24 cubic units How do we find its volume in cubic inches 1 We know that each cube with a 12 inch edge length has a volume of 8 cubic inch 1 because 12 2 12 18 Since the prism is built using 24 of these cubes its volume in cubic inches would then be 24 18 or 3 cubic inches The volume of the prism in cubic inches can also be found by multiplying the fractional edge lengths in inches 1 12 2 4 3 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 G6M4 LESSON 14 Name Date GRADE 6 MISSION 4 LESSON 14 Exit Ticket 2 5 1 1 A triangle has a base of 3 Show your reasoning inches and an area of 5 10 square inches Find the height of the triangle 2 Answer each of the following questions and show your reasoning a How many cubes with an edge length of length of 1 inch 1 3 inch are needed to build a cube with an edge b What is the volume in cubic inches of one cube with an edge length of 1 3 inch 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M4 LESSON 15 Lesson 15 Volume of Prisms Let s look at the volume of prisms that have fractional measurements Warm Up 1 1 Answer the questions about the prism How many cubes with an edge length of 1 inch fill this box 4 in 3 in 10 in 2 If the cubes had an edge length of 2 inches would more or fewer cubes be needed to fill the box Explain how you know 3 If the cubes had an edge length of Explain how you know 1 2 inch would more or fewer cubes be needed to fill the box 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 2 2 Answer the questions about rectangular prisms with fractional edge lengths 1 Diego correctly points out that 108 cubes with an edge length of 13 inch are needed to fill a rectangular prism that is 3 inches by 1 inch by 1 13 inch Explain how this is true Draw a sketch if needed 2 What is the volume in cubic inches of the rectangular prism Show your reasoning 3 Lin and Noah are packing small cubes into a cube with an edge length of 1 12 inches Lin is using cubes with an edge length of 12 inch and Noah is using cubes with an edge length of 14 inch a Who would need more cubes to fill the 1 12 inch cube Show how you know b If Lin and Noah use their small cubes to find the volume of the 1 12 inch cube would they get the same value Explain your reasoning 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 G6M4 LESSON 15 Lesson Summary If a rectangular prism has edge lengths a units b units and c units the volume is the product of a b and c V a b c This means that if we know the volume and two edge lengths we can divide to find the third edge length Suppose the volume of a rectangular prism is 400 12 cm3 one edge length is 11 2 cm another is 6 cm and the third edge length is unknown We can write a multiplication equation to represent the situation 11 2 6 400 1 2 We can find the third edge length by dividing 400 1 2 11 2 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 97

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 15 Date GRADE 6 MISSION 4 LESSON 15 Exit Ticket A storage box has a volume of 56 cubic inches and the base of the box is 4 inches by 4 inches 1 What is the height of the box 2 Lin s teacher uses the box to store her set of cubes with an edge length of completely full how many cubes are in the set 1 2 inch If the box is 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 99

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ZEARN MATH STUDENT EDITION G6M4 LESSON 16 Lesson 16 Solving Problems Involving Fractions Let s add subtract multiply and divide fractions Warm Up 1 Without calculating order the expressions according to their values from least to greatest Be prepared to explain your reasoning 3 4 2 3 3 4 2 3 3 4 2 3 3 4 2 3 Concept Exploration ACTIVITY 1 2 Work with a partner to write equations for the questions in your notes One person should work on the questions labeled A1 B1 E1 and the other should work on those labeled A2 B2 E2 1 A1 Lin s bottle holds 3 4 cups of water She drank 1 cup of water What fraction of the water in the bottle did she drink B1 Plant A is 16 3 feet tall This is Plant B How tall is Plant B 4 5 as tall as 1 A2 Lin s bottle holds 3 4 cups of water After 1 she drank some there were 1 2 cups of water in the bottle How many cups did she drink 16 B2 Plant A is 3 feet tall Plant C is Plant A How tall is Plant C 4 5 as tall as 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 16 ZEARN MATH STUDENT EDITION C1 89 kilogram of berries is put into a container 7 that already has 3 kilograms of berries How many kilograms are in the container 1 D1 The area of a rectangle is 14 2 sq cm and one 1 side is 4 2 cm How long is the other side 2 E1 A stack of magazines is 4 5 inches high The 1 stack needs to fit into a box that is 2 8 inches high How many inches too high is the stack 8 C2 A container with 9 kilogram of berries is 2 3 full How many kilograms can the container hold 1 D2 The side lengths of a rectangle are 4 2 cm 2 and 2 5 cm What is the area of the rectangle 2 E2 A stack of magazines is 4 5 inches high 2 Each magazine is 5 inch thick How many magazines are in the stack 2 Trade papers with your partner and check your partner s equations If there is a disagreement about what an equation should be discuss it until you reach an agreement 3 Your teacher will assign 2 3 questions for you to answer For each question a Estimate the answer before calculating it b Find the answer and show your reasoning 102 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M4 LESSON 16 Lesson Summary We can add subtract multiply and divide both whole numbers and fractions Here is a summary of how we add subtract multiply and divide fractions To add or subtract fractions we often look for a common denominator so the pieces involved are the same size This makes it easy to add or subtract the pieces To multiply fractions we often multiply the numerators and the denominators To divide a number by a fraction b we can simply b multiply the number by a which is the reciprocal of 3 2 3 8 4 7 4 5 5 9 8 15 10 10 3 5 8 9 a a b 5 3 4 7 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 103

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ZEARN MATH STUDENT EDITION Name G6M4 LESSON 16 Date GRADE 6 MISSION 4 LESSON 16 Exit Ticket 1 5 A box of pencils is 5 4 inches wide Seven pencils laid side by side take up 2 8 inches of the width 1 How many inches of the width of the box is not taken up by pencils Show your reasoning 2 All the pencils have the same width How wide is each pencil 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 105

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ZEARN MATH STUDENT EDITION G6M4 LESSON 17 Lesson 17 Fitting Boxes into Boxes Let s use what we learned about fractions to find shipping costs Warm Up 1 An artist makes necklaces She packs each necklace in a small jewelry box that is 1 34 inches by 2 14 inches by 34 inch A department store ordered 270 necklaces The artist plans to ship the necklaces to the department store using flat rate shipping boxes from the post office Which of the flat rate boxes should she use to minimize her shipping cost 1 Read the problem statement What additional information will you need to solve this problem 2 Discuss this information with your group Make a plan for using this information to find the most inexpensive way to ship the jewelry boxes Once you have agreed on a plan write down the main steps 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M4 LESSON 17 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Work with your group to find the best plan for shipping the boxes of necklaces Each member of your group should select a different type of flat rate shipping box and answer the following questions Recall that each jewelry box is 1 to be shipped 3 4 inches by 2 1 4 inches by 3 4 inch and that there are 270 jewelry boxes USPS flat rate information 3 8 Small box 5 Medium box 1 11 inches by 8 inches by 8 5 8 Medium box 2 11 7 8 inches by 1 1 2 inches by 3 5 8 inches by 5 3 8 inches Cost 6 80 1 2 inches Cost 13 45 inches by 13 Large box 12 inches by 12 inches by 5 1 2 5 8 inches Cost 13 45 inches Cost 18 75 For each type of flat rate shipping box 108 1 Find how many jewelry boxes can fit into the box Explain or show how the jewelry boxes can be packed in the shipping box Draw a sketch to show your thinking if needed 2 Calculate the total cost of shipping all 270 jewelry boxes in shipping boxes of that type Show your reasoning and 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 G6M4 LESSON 17 ACTIVITY 2 3 Discuss the following questions as a group 1 Share and discuss your work with the other members of your group Your teacher will display questions to guide your discussion Note the feedback from your group so you can use it to revise your work 2 Using the feedback from your group revise your work to improve its correctness clarity and accuracy Correct any errors You may also want to add notes or diagrams or remove unnecessary information 3 Which shipping boxes should the artist use As a group decide which boxes you recommend for shipping 270 jewelry boxes Be prepared to share 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 109

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Grade 6 Mission 5 Arithmetic in Base Ten

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ZEARN MATH STUDENT EDITION G6M5 LESSON 1 3 Lesson 3 Using Decimals in a Shopping Context Let s use what we know about decimals to make shopping decisions Warm Up 1 Clare went to a concession stand that sells pretzels for 3 25 drinks for 1 85 and bags of popcorn for 0 99 each She bought at least one of each item and spent no more than 10 1 Could Clare have purchased 2 pretzels 2 drinks and 2 bags of popcorn Explain your reasoning 2 Could she have bought 1 pretzel 1 drink and 5 bags of popcorn Explain your reasoning Concept Exploration ACTIVITY 1 2 You are planning a dinner party with a budget of 50 and a menu that consists of 1 main dish 2 side dishes and 1 dessert There will be 8 guests at your party Choose your menu items and decide on the quantities to buy so you stay on budget If you choose meat fish or poultry for your main dish plan to buy at least 0 5 pound per person Use the worksheet to record your choices and estimated costs Then find the estimated total cost and cost per person See examples in the first two rows 1 The budget is per guest 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 3 Item ZEARN MATH STUDENT EDITION Quantity Needed Advertised Price Estimated Subtotal in dollars Ex Main Dish Fish 4 pounds 6 69 per pound 4 7 28 Ex Dessert Cupcakes 8 cupcakes 2 99 per 6 cupcakes 2 3 6 Estimated cost per person in dollars Main Dish Side Dish 1 Side Dish 2 Dessert Estimated Total 114 2 Is your estimated total close to your budget If so continue to the next question If not revise your menu choices until your estimated total is close to the budget 3 Calculate the actual costs of the two most expensive items and add them Show your reasoning 4 How will you know if your total cost for all menu items will or will not exceed your budget Is there a way to predict this without adding all the exact costs 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 G6M5 LESSON 3 Lesson Summary We often use decimals when dealing with money In these situations sometimes we round and make estimates and other times we calculate the numbers more precisely There are many different ways we can add subtract multiply and divide decimals When we perform these computations it is helpful to understand the meanings of the digits in a number and the properties of operations We will investigate how these understandings help us work with decimals in upcoming lessons 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 115

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ZEARN MATH STUDENT EDITION Name G6M5 LESSON 3 Date GRADE 6 MISSION 5 LESSON 3 Exit Ticket Planning your menu involved many calculations with decimals Reflect on how you made these calculations 1 How did you compute sums of dollar amounts that were not whole numbers For example how did you compute the sum of 5 89 and 1 45 Use this example to explain your strategy 2 How did you compute products of dollar amounts that were not whole numbers For example how did you compute the cost of 4 pounds of beef at 5 89 per pound Use this example to explain your strategy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 2 Lesson 2 Using Diagrams to Represent Addition and Subtraction Let s represent addition and subtraction of decimals Warm Up 1 Here is a rectangle What number does the rectangle represent if each small square represents a 1 b 0 1 c 0 01 d 0 001 2 Here is a square What number does the square represent if each small rectangle represents a 10 b 0 1 c 0 00001 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 Here are two ways to calculate the value of 0 26 0 07 In the diagram each rectangle represents 0 1 and each square represents 0 01 1 0 26 0 07 0 33 Bundle Use what you know about base ten units and addition of base ten numbers to explain a Why ten squares can be bundled into a rectangle b How this bundling is reflected in the computation 43 1 120 Use place value diagrams to help you solve the following problems Find the value of 0 38 0 69 by drawing a diagram Can you find the sum without bundling Would it be useful to bundle some pieces 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 G6M5 LESSON 2 2 Calculate 0 38 0 69 Check your calculation against your diagram in the previous question 3 Find each sum The larger square represents 1 the rectangle represents 0 1 and the smaller square represents 0 01 a b 6 03 0 098 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary Base ten diagrams represent collections of base ten units tens ones tenths hundredths etc We can use them to help us understand sums of decimals Here is a diagram of 0 008 and 0 013 where a square represents 0 001 and a rectangle made up of ten squares represents 0 01 hundredths thousandths 0 008 0 013 To find the sum we can bundle or compose 10 thousandths as 1 hundredth hundredths thousandths 0 008 0 013 Bundle Here is a diagram of the sum which shows 2 hundredths and 1 thousandth 0 021 We can use vertical calculation to find Notice that here 10 thousandths are also bundled or composed as 1 hundredth 1 0 008 0 013 0 021 122 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 G6M5 LESSON 2 Name Date GRADE 6 MISSION 5 LESSON 2 Exit Ticket Is this equation true 0 025 0 17 0 042 Use a diagram or numerical calculation to explain or show your reasoning Here are diagrams that you could use to represent base ten units 0 1 tenth 1 one 0 01 hundredth 0 001 thousandth 0 0001 ten thousandth 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 3 Lesson 3 Adding and Subtracting Decimals with Few Non Zero Digits Let s add and subtract decimals Warm Up 1 2 Evaluate mentally 1 009 0 391 Decide if each equation is true or false Be prepared to explain your reasoning a 34 56000 34 56 b 25 25 0 c 2 405 2 45 Concept Exploration ACTIVITY 1 3 To represent 0 4 0 03 Diego and Noah drew different diagrams Each rectangle shown here represents 0 1 Each square represents 0 01 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 3 ZEARN MATH STUDENT EDITION Diego started by drawing 4 rectangles for 0 4 He then replaced 1 rectangle with 10 squares and crossed out 3 squares for the subtraction of 0 03 leaving 3 rectangles and 7 squares in his drawing tenths hundredths unbundle Diego s method Noah started by drawing 4 rectangles for 0 4 He then crossed out 3 of them to represent the subtraction leaving 1 rectangle in his drawing tenths Noah s method Do you agree that either diagram correctly represents 0 4 0 03 Discuss your reasoning with a partner 43 To represent 0 4 0 03 Elena drew another diagram She also started by drawing 4 rectangles She then replaced all 4 rectangles with 40 squares and crossed out 3 squares for the subtraction of 0 03 leaving 37 squares in her drawing Is her diagram correct Discuss your reasoning with a partner tenths hundredths unbundle Elena s method 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 53 G6M5 LESSON 3 Find each difference Explain or show your reasoning a 0 3 0 05 b 2 1 0 4 c 1 03 0 06 d 0 02 0 007 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary Base ten diagrams can help us understand subtraction as well as addition Suppose we are finding 0 023 0 007 Here is a diagram showing 0 023 or 2 hundredths and 3 thousandths hundredths thousandths 0 023 Subtracting 7 thousandths means removing 7 small squares but we do not have enough to remove Because 1 hundredth is equal to 10 thousandths we can unbundle or decompose one of the hundredths 1 rectangle into 10 thousandths 10 small squares hundredths thousandths 0 023 unbundle We now have 1 hundredth and 13 thousandths from which we can remove 7 thousandths hundredths thousandths 0 023 subtract 0 07 We have 1 hundredth and 6 thousandths remaining so 0 023 0 007 0 016 hundredths thousandths 0 016 Here is a vertical calculation of 0 023 0 007 1 13 0 023 0 007 0 016 In both calculations notice that a hundredth is unbundled or decomposed into 13 thousandths in order to subtract 7 thousandths 128 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 3 Date GRADE 6 MISSION 5 LESSON 3 Exit Ticket 1 Find the sum 1 56 0 083 Show your reasoning 2 Find the difference 0 2 0 05 Show your reasoning 3 You need to be at least 39 37 inches tall about a meter to ride on a bumper car Diego s cousin is 35 75 inches tall How many more inches will he need to grow before Diego can take him on the bumper car ride 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 129

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ZEARN MATH STUDENT EDITION G6M5 LESSON 4 Lesson 4 Adding and Subtracting Decimals with Many Non Zero Digits Let s practice adding and subtracting decimals Warm Up 1 Clare bought a photo for 17 cents and paid with a 5 bill Answer the following questions about this situation 5 0 17 5 0 17 5 0 17 1 Which way of writing the numbers could Clare use to find the change she should receive Be prepared to explain how you know 2 Find the amount of change that Clare should receive Show your reasoning and be prepared to explain how you calculate the difference of 0 17 and 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 131

G6M5 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Solve the following subtraction problems Find the value of each expression Show your reasoning a 11 3 9 5 b 318 8 94 63 c 2 Discuss with a partner Which method or methods did you use in the previous question Why In what ways were your methods effective Was there an expression for which your methods did not work as well as expected 3 1 132 0 02 0 0116 For each situation write an equation and use it to solve Lin s grandmother ordered needles that were 0 3125 inch long to administer her medication but the pharmacist sent her needles that were 0 6875 inch long How much longer were these needles than the ones she ordered Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M5 LESSON 4 2 There is 0 162 liter of water in a 1 liter bottle How much more water should be put in the bottle so it contains exactly 1 liter Show your reasoning 3 One micrometer is 1 millionth of a meter A red blood cell is about 7 5 micrometers in diameter A coarse grain of sand is about 70 micrometers in diameter Find the difference between the two diameters in meters Show your reasoning ACTIVITY 2 Write the missing digits in each calculation so that the value of each sum or difference is correct Be prepared to explain your reasoning 43 1 3 5 0 404 2 1 0 7 0 012 4 9 8765 10 7 3 4567 70 0 0089 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary Base ten diagrams work best for representing subtraction of numbers with few non zero digits such as 0 16 0 09 For numbers with many non zero digits such as 0 25103 0 04671 it would take a long time to draw the base ten diagram With vertical calculations we can find this difference efficiently Thinking about base ten diagrams can help us make sense of this calculation 10 4 0 10 0 25103 0 04671 0 20432 The thousandth in 0 25103 is unbundled or decomposed to make 10 ten thousandths so that we can subtract 7 ten thousandths Similarly one of the hundredths in 0 25103 is unbundled or decomposed to make 10 thousandths 134 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 4 Date GRADE 6 MISSION 5 LESSON 4 Exit Ticket 1 Diego is 59 5 inches tall His brother is 40 125 inches tall How much taller than his brother is Diego Show your reasoning 2 A runner has run 1 192 kilometers of a 10 kilometer race How much farther does he need to run to finish the race 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 135

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ZEARN MATH STUDENT EDITION G6M5 LESSON 5 Lesson 5 Decimal Points in Products Let s look at products that are decimals Warm Up 1 In which equation is the value of x the largest x 10 810 2 x 10 81 x 10 8 1 x 10 0 81 How many times the size of 0 81 is 810 Concept Exploration ACTIVITY 1 3 1 Work with a partner to answer the following questions One person should answer the questions labeled Partner A and the other should answer those labeled Partner B Then compare the results Find each product or quotient Be prepared to explain your reasoning Partner A Partner B 1 a 250 10 b 250 100 1 b 250 100 c c a 250 10 48 10 d 48 100 1 48 10 1 d 48 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 137

G6M5 LESSON 5 ZEARN MATH STUDENT EDITION a Use your work in the previous problems to find 720 0 1 and 720 0 01 Explain your reasoning 43 a 36 0 1 d 54 0 01 b 24 5 0 1 e 9 2 0 01 c 53 138 Find each product Show your reasoning 1 8 0 1 Jada says If you multiply a number by 0 001 the decimal point of the number moves three places to the left Do you agree with her statement 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 G6M5 LESSON 5 ACTIVITY 2 63 1 Answer the following questions about multiplying decimals and fractions Select all expressions that are equivalent to 0 6 0 5 Be prepared to explain your reasoning a 6 0 1 5 0 1 e 6 0 001 5 0 01 b 6 0 01 5 0 1 f 6 5 10 10 g 6 10 c 1 1 6 10 5 10 1 1 1 5 10 1 d 6 1000 5 100 2 Find the value of 0 6 0 5 Show your reasoning 3 Find the value of each product by writing and reasoning with an equivalent expression with fractions a 0 3 0 02 b 0 7 0 05 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 5 ZEARN MATH STUDENT EDITION Lesson Summary 1 1 We can use fractions like 10 and 100 to reason about the location of the decimal point in a product of two decimals Let s take 24 0 1 as an example There are several ways to find the product We can interpret it as 24 groups of 1 tenth or 24 tenths which is 2 4 We can think of it as 24 10 which is equal to 10 and also equal to 2 4 Multiplying by 10 has the same result as dividing by 10 so we can also think of the product as 24 10 which is equal to 2 4 1 24 1 Similarly we can think of 0 7 0 09 as 7 tenths times 9 hundredths and write 1 1 7 10 9 100 We can rearrange whole numbers and fractions 1 1 1 63 7 9 10 100 63 1000 1000 This tells us that 0 7 0 09 0 063 Here is another example To find 1 5 0 43 we can think of 1 5 as 15 tenths and 0 43 as 43 hundredths We can write the tenths and hundredths as fractions and rearrange the factors 1 1 1 1 Multiplying 15 and 43 gives us 645 and multiplying 10 and 100 gives us 1000 So 1 5 0 43 is 645 1000 which is 0 645 140 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 Name G6M5 LESSON 5 Date GRADE 6 MISSION 5 LESSON 5 Exit Ticket 1 Use what you know about decimals or fractions to explain why 0 2 0 002 0 0004 2 A rectangular plot of land is 0 4 kilometer long and 0 07 kilometer wide What is its area in square kilometers 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 141

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ZEARN MATH STUDENT EDITION G6M5 LESSON 6 Lesson 6 Methods for Multiplying Decimals Let s look at some ways we can represent multiplication of decimals Warm Up 1 Which expression doesn t belong Explain your reasoning a 2 0 3 b 2 3 0 1 c 6 0 1 d 0 1 6 Concept Exploration ACTIVITY 1 2 Elena and Noah used different methods to compute 0 23 1 5 Both computations were correct 0 23 100 23 1 5 10 15 23 15 345 345 1 000 0 345 23 0 23 100 1 5 15 10 23 15 345 100 100 1 000 345 1 000 0 345 Elena s method Noah s method Analyze the two methods then discuss these questions with your partner Which method makes more sense to you Why What might Elena do to compute 0 16 0 03 What might Noah do to compute 0 16 0 03 Will the two methods result in the same value 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 6 3 ZEARN MATH STUDENT EDITION Compute each product using the equation 21 47 987 and what you know about fractions decimals and place value Explain your reasoning a 2 1 4 7 b 21 0 047 c 0 021 4 7 Lesson Summary Here are three other ways to calculate a product of two decimals such as 0 04 0 07 First we can multiply each decimal by the same power of 10 to obtain whole number factors Because we multiplied both 0 04 and 0 07 by 100 to get 4 and 7 the product 28 is 100 100 times the original product so we need to divide 28 by 10 000 0 04 100 4 0 07 100 7 4 7 28 28 10 000 0 0028 4 7 Second we can write each decimal as a fraction 0 04 100 and 0 07 100 and multiply them 4 100 7 100 28 10 000 0 0028 Third we can use an area model The product 0 04 0 07 can be thought of as the area of a rectangle with side lengths of 0 04 unit and 0 07 unit 0 04 0 07 1 In this diagram each small square is 0 01 unit by 0 01 unit Its area in square units is therefore 100 1 1 100 which is 10 000 Because the rectangle is composed of 28 small squares its area in square units must be 28 1 10 000 28 10 000 0 0028 All three calculations show that 0 04 0 07 0 0028 144 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 6 Date GRADE 6 MISSION 5 LESSON 6 Exit Ticket 1 Use the equation 135 42 5 670 and what you know about fractions decimals and place value to explain how to place the decimal point when you compute 1 35 4 2 2 Which of the following is the correct value of 0 22 0 4 Show your reasoning a 8 8 b 0 88 c 0 088 d 0 0088 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G6M5 LESSON 7 Lesson 7 Using Diagrams to Represent Multiplication Let s use area diagrams to find products Warm Up 1 1 For each of the following proucts choose the best estimate of its value Be prepared to explain your reasoning 6 8 2 3 a 1 40 b 14 c 140 2 74 8 1 a 5 6 b 56 c 560 3 166 0 09 a 1 66 b 16 6 c 166 4 3 4 1 9 a 6 5 b 65 c 650 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 You can use area diagrams to represent products of decimals Here is an area diagram represents 2 4 1 3 2 0 4 1 0 3 a Which region represents 0 4 0 3 Label that region with its area of 0 12 b Label each of the other regions with their respective areas c Find the value of 2 4 1 3 Show your reasoning 3 Here are two ways of calculating 2 4 1 3 Analyze the calculations and discuss the following questions with a partner 2 4 1 3 0 1 2 0 6 0 4 2 3 1 2 2 4 1 3 0 7 2 2 4 3 1 2 partial products Calculation A Calculation B Analyze the calculations and discuss with a partner a Which two numbers are being multiplied to get 0 12 in Calculation A Which numbers are being multiplied to get 0 72 in Calculation B How are the other numbers in blue calculated b In each calculation why are the numbers in grey lined up vertically the way they are 148 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 43 53 G6M5 LESSON 7 Find the product of 3 1 1 5 by drawing and labeling an area diagram Show your reasoning Show how to calculate 3 1 1 5 using numbers without a diagram Be prepared to explain your reasoning If you are stuck use the examples in a previous question to help you Lesson Summary Suppose that we want to calculate the product of two numbers that are written in base ten To explain how we can use what we know about base ten numbers and areas of rectangles Here is a diagram of a rectangle whose side lengths are 3 4 units and 1 2 units Its area in square units is the product 3 4 1 2 To calculate this product and find the area of the rectangle we can decompose each side length into its base ten units 3 4 3 0 4 and 1 2 1 0 2 decomposing the rectangle into four smaller sub rectangles 3 0 4 1 D C 0 2 B A We can rewrite the product and expand it twice 3 4 1 2 3 0 4 1 0 2 3 0 4 1 3 0 4 0 2 3 1 3 0 2 0 4 1 0 4 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 149

G6M5 LESSON 7 ZEARN MATH STUDENT EDITION In the last expression each of the four terms is called a partial product Each partial product gives the area of a sub rectangle in the diagram The sum of the four partial products gives the area of the entire rectangle We can show the horizontal calculations above as two vertical calculations 3 4 1 2 1 0 0 8 0 6 0 4 3 4 0 8 3 4 1 2 1 0 6 8 3 4 4 0 8 A B C D A C B D The vertical calculation on the left is an example of the partial products method It shows the values of each partial product and the letter of the corresponding sub rectangle Each partial product gives an area A is 0 2 unit by 0 4 unit so its area is 0 08 square unit B is 3 units by 0 2 unit so its area is 0 6 square unit C is 0 4 unit by 1 unit so its area is 0 4 square unit D is 3 units by 1 unit so its area is 3 square units The sum of the partial products is 0 08 0 6 0 4 3 so the area of the rectangle is 4 08 square units The calculation on the right shows the values of two products Each value gives a combined area of two subsrectangles 150 The combined regions of A and B have an area of 0 68 square units 0 68 is the value of 3 0 4 0 2 The combined regions of C and D have an area of 3 4 square units 3 4 is the value of 3 0 4 1 The sum of the values of two products is 0 68 3 4 so the area of the rectangle is 4 08 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

ZEARN MATH STUDENT EDITION Name G6M LESSON 7 Date GRADE 6 MISSION 5 LESSON 7 Exit Ticket Find 4 2 1 6 by drawing an area diagram or using another method 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 151

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ZEARN MATH STUDENT EDITION G6M5 LESSON 8 Lesson 8 Calculating Products of Decimals Let s multiply decimals Warm Up 1 Evaluate mentally 20 5 20 0 8 20 0 04 20 5 84 Concept Exploration ACTIVITY 1 2 A common way to find a product of decimals is to calculate a product of whole numbers then place the decimal point in the product Here is an example for 2 5 1 2 Use what you know about decimals and place value to explain why the decimal point of the product is placed where it is 25 12 50 250 300 25 12 300 2 5 1 2 3 00 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 8 3 ZEARN MATH STUDENT EDITION Use the method shown in the first question to calculate each product a 4 6 0 9 b 16 5 0 7 154 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 43 G6M5 LESSON 8 Use area diagrams to check your earlier calculations For each problem Decompose each number into its base ten units and write them in the boxes on each side of the rectangle Write the area of each lettered region in the diagram Then find the area of the entire rectangle Show your reasoning a 4 6 0 9 B A b 16 5 0 7 C 53 B A About how many centimeters are in 6 25 inches if 1 inch is about 2 5 centimeters 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 155

G6M5 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary We can use 84 43 and what we know about place value to find 8 4 4 3 Since 8 4 is 84 tenths and 4 3 is 43 tenths then 43 8 4 4 3 84 10 10 8 4 4 3 84 43 100 That means we can compute and then divide by 100 to find 8 4 4 3 84 43 3612 8 4 4 3 36 12 1 1 1 Using fractions such as 10 100 and 1 000 allows us to find the product of two decimals using the following steps Write each decimal factor as a product of a whole number and a fraction Multiply the whole numbers Multiply the fractions Multiply the products of the whole numbers and fractions 1 1 1 We know multiplying by fractions such as 10 100 and 1 000 is the same as dividing by 10 100 and 1 000 respectively This means we can move the decimal point in the whole number product to the left the appropriate number of spaces to correctly place the decimal point 156 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 8 Date GRADE 6 MISSION 5 LESSON 8 Exit Ticket Calculate 1 6 0 215 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 157

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ZEARN MATH STUDENT EDITION G6M5 LESSON 9 Lesson 9 Using the Partial Quotients Method Let s divide whole numbers Warm Up 1 Elena used base ten diagrams to find 372 3 She started by representing 372 She made 3 groups each with 1 hundred Then she put the tens and ones in each of the 3 groups to create a diagram for 372 3 3 hundreds hundreds 7 tens tens 2 ones ones Discuss with a partner Elena s diagram for 372 has 7 tens The one for 372 3 has only 6 tens Why Where did the extra ones small squares come from 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 159

G6M5 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Andre calculated 657 3 using a method that was different from Elena s 2 Andre s method He started by writing the dividend 657 and the divisor 3 3 657 1 He then subtracted 3 groups of different amounts from 657 starting with 3 groups of 200 then 3 groups of 10 and then 3 groups of 9 200 3 657 600 57 9 10 200 3 657 600 57 30 27 27 0 Andre calculated 200 10 9 and then wrote 219 219 9 10 200 3 657 600 57 30 27 27 0 Discuss the following questions with a partner a Andre subtracted 600 from 657 What does the 600 represent b Andre wrote 10 above the 200 and then subtracted 30 from 57 How is the 30 related to the 10 c What do the numbers 200 10 and 9 represent d What is the meaning of the 0 at the bottom of Andre s work 2 How might Andre calculate 896 4 Explain or show your reasoning ACTIVITY 2 160 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 3 G6M5 LESSON 9 Solve the following problems Show your reasoning 1 Find the quotient of 1 332 9 using one of the methods you have seen so far Show your reasoning 2 Find each quotient and show your reasoning Use the partial quotients method at least once a 1 115 5 b 665 7 c 432 16 Lesson Summary We can find the quotient for 345 3 in different ways One way is to use a base ten diagram to represent the hundreds tens and ones and to create equal sized groups hundreds tens ones 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 9 ZEARN MATH STUDENT EDITION We can think of the division by 3 as splitting up 345 into 3 equal groups hundreds tens ones group 1 group 2 group 3 Each group has 1 hundred 1 ten and 5 ones so 345 3 115 Notice that in order to split 345 into 3 equal groups one of the tens had to be unbundled or decomposed into 10 ones Another way to divide 345 by 3 is by using the partial quotients method in which we keep subtracting 3 groups of some amount from 345 115 5 10 100 3 345 300 45 30 15 15 0 3 groups of 100 3 groups of 10 3 groups of 5 115 50 50 15 3 345 45 300 150 150 150 0 3 groups of 15 3 groups of 50 3 groups of 50 In the calculation on the left first we subtract 3 groups of 100 then 3 groups of 10 and then 3 groups of 5 Adding up the partial quotients 100 10 5 gives us 115 The calculation on the right shows a different amount per group subtracted each time 3 groups of 15 3 groups of 50 and 3 more groups of 50 but the total amount in each of the 3 groups is still 115 There are other ways of calculating 345 3 using the partial quotients method Both the base ten diagrams and partial quotients methods are effective If however the dividend and divisor are large as in 1 248 26 then the base ten diagrams will be time consuming 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 Name G6M5 LESSON 9 Date GRADE 6 MISSION 5 LESSON 9 Exit Ticket Calculate 4 235 11 using any method 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

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ZEARN MATH STUDENT EDITION G6M5 LESSON 10 Lesson 10 Using Long Division Let s use long division Warm Up 1 500 7 2 Estimate the quotient mentally Do another one Estimate the quotient mentally 1 394 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 165

G6M5 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 Lin has a method of calculating quotients that is different from Elena s method and Andre s method Here is how she found the quotient of 657 3 Lin arranged the numbers for vertical calculations Her plan was to divide each digit of 657 into 3 groups starting with the 6 hundreds 3 657 There are 3 groups of 2 in 6 so Lin wrote 2 at the top and subtracted 6 from the 6 leaving 0 Then she brought down the 5 tens of 657 2 3 657 6 05 There are 3 groups of 1 in 5 so she wrote 1 at the top and subtracted 3 from 5 which left a remainder of 2 21 3 657 6 5 3 2 She brought down the 7 ones of 657 and wrote it next to the 2 which made 27 There are 3 groups of 9 in 27 so she wrote 9 at the top and subtracted 27 leaving 0 219 3 657 6 5 3 27 27 0 Discuss with your partner how Lin s method is similar to and different from drawing base ten diagrams or using the partial quotients method 166 Lin subtracted 3 2 then 3 1 and lastly 3 9 Earlier Andre subtracted 3 200 then 3 10 and lastly 3 9 Why did they have the same quotient In the third step why do you think Lin wrote the 7 next to the remainder of 2 rather than adding 7 and 2 to get 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 43 G6M5 LESSON 10 Lin s method is called long division Use this method to find the following quotients Check your answer by multiplying it by the divisor a 846 3 b 1 816 4 c 768 12 Lesson Summary Long division is another method for calculating quotients It relies on place value to perform and record the division When we use long division we work from left to right and with one digit at a time starting with the leftmost digit of the dividend We remove the largest group possible each time using the placement of the digit to indicate the size of each group Here is an example of how to find 948 3 using long division 316 3 948 9 4 3 18 18 0 3 groups of 3 hundreds 3 groups of 1 ten 3 groups of 6 ones We start by dividing 9 hundreds into 3 groups which means 3 hundreds in each group Instead of writing 300 we simply write 3 in the hundreds place knowing that it means 3 hundreds There are no remaining hundreds so we work with the tens We can make 3 groups of 1 ten in 4 tens so we write 1 in the tens place above the 4 of 948 Subtracting 3 tens from 4 tens we have a remainder of 1 ten We know that 1 ten is 10 ones Combining these with the 8 ones from 948 we have 18 ones We can make 3 groups of 6 so we write 6 in the ones place In total there are 3 groups of 3 hundreds 1 ten and 6 ones in 948 so 948 3 316 TERMINOLOGY Long division 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 167

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ZEARN MATH STUDENT EDITION Name G6M5 LESSON 10 Date GRADE 6 MISSION 5 LESSON 10 Exit Ticket Use long division to find the value of 1 875 15 Show your work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 169

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ZEARN MATH STUDENT EDITION G6M5 LESSON 11 Lesson 11 Dividing Numbers that Result in Decimals Let s find quotients that are not whole numbers Warm Up 1 Find the quotients mentally 400 8 80 8 16 8 496 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here is how Lin calculated 62 5 Lin set up the numbers for long division 5 62 She subtracted 5 times 1 from the 6 which leaves a remainder of 1 Lin drew a vertical line and a decimal point separating the ones and tenths place Lastly she subtracted 5 times 4 from 20 which left no remainder She wrote the 2 from 62 next to the 1 which made 12 and subtracted 5 times 2 from the 12 12 10 is 2 She wrote 0 to the right of 2 which made 20 At the top she wrote 4 next to the decimal point 12 5 62 5 12 10 20 12 4 5 62 5 12 10 20 20 0 12 5 62 5 12 10 2 Discuss with your partner 172 Lin put a 0 after the remainder of 2 Why Why does this 0 not change the value of the quotient Lin subtracted 5 groups of 4 from 20 What value does the 4 in the quotient represent What value did Lin find for 62 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

ZEARN MATH STUDENT EDITION 3 1 G6M5 LESSON 11 Use your understanding of long division to solve the following problems Use long division to find the value of each expression a 126 8 b 90 12 2 Use long division to show that a 5 4 or 5 4 is 1 25 b 4 5 or 4 5 is 0 8 c 1 8 is 0 125 1 8 or 1 d 1 25 or 25 is 0 04 3 Noah said we cannot use long division to calculate 10 3 because there will always be a remainder a What do you think Noah meant by there will always be a remainder b Do you agree with his statement Why or why not 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 173

G6M5 LESSON 11 ZEARN MATH STUDENT EDITION Lesson Summary Dividing a whole number by another whole number does not always produce a whole number quotient Let s look at 86 4 which we can think of as dividing 86 into 4 equal groups tens ones unbundle tenths We can see in the base ten diagram that there are 4 groups of 21 in 86 with 2 ones left over To find the quotient we need to distribute the 2 ones into the 4 groups To do this we can unbundle or decompose the 2 ones into 20 tenths which enables us to put 5 tenths in each group Once the 20 tenths are distributed each group will have 2 tens 1 one and 5 tenths so 86 4 21 5 21 5 4 86 8 6 4 20 20 0 174 We can also calculate 86 4 using long division The calculation shows that after removing 4 groups of 21 there are 2 ones remaining We can continue dividing by writing a 0 to the right of the 2 and thinking of that remainder as 20 tenths which can then be divided into 4 groups To show that the quotient we are working with now is in the tenth place we put a decimal point to the right of the 1 which is in the ones place at the top It may also be helpful to draw a vertical line to separate the ones and the tenths There are 4 groups of 5 tenths in 20 tenths so we write 5 in the tenths place at the top The calculation likewise shows 86 4 21 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

ZEARN MATH STUDENT EDITION Name G6M5 LESSON 11 Date GRADE 6 MISSION 5 LESSON 11 Exit Ticket Use long division to find each quotient Show your computation and write your answer as a decimal 1 22 5 2 7 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 12 Lesson 12 Dividing Decimals by Whole Numbers Let s divide decimals by whole numbers Warm Up 1 Find the quotients mentally 80 4 12 4 1 2 4 81 2 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 177

G6M5 LESSON 12 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 LAUNCH Elena is finding 53 8 4 using diagrams Elena began by representing 53 8 5 tens 3 ones 8 tenths She placed 1 ten into each group unbundled the remaining 1 ten into 10 ones and went on distributing the units This diagram shows Elena s initial placement of the units and the unbundling of 1 ten Tens Ones Tenths Hundredths Group 1 Group 2 Group 3 Group 4 Unbundle 178 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M5 LESSON 12 ACTIVITY 1 2 1 Use your understanding of division to solve problems 1 4 in your notes Complete the diagram by continuing the division process How would you use the available units to make 4 equal groups As the units get placed into groups show them accordingly and cross out those pieces from the bottom If you unbundle a unit draw the resulting pieces 2 What value did you find for 53 8 4 Be prepared to explain your reasoning 3 Use long division to find 53 8 4 Check your answer by multiplying it by the divisor 4 4 Use long division to find 77 4 5 If you get stuck you can draw diagrams or use another method ACTIVITY 2 3 Analyze the dividends divisors and quotients in the calculations then answer the questions 24 3 72 6 12 12 0 1 24 30 720 60 120 120 0 24 300 7200 600 1200 1200 0 24 3000 72000 6000 12000 12000 0 Complete each sentence In the calculations above Each dividend is ____________ times the dividend to the left of it Each divisor is ____________ times the divisor to the left of it Each quotient is ____________ the quotient to the left of 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 179

G6M5 LESSON 12 2 ZEARN MATH STUDENT EDITION Suppose we are writing a calculation to the right of 72 000 3 000 Which expression has a quotient of 24 Be prepared to explain your reasoning a 72 000 30 000 c 720 000 30 000 b 720 000 300 000 d 720 000 3 000 3 Suppose we are writing a calculation to the left of 72 3 Write an expression that would also give a quotient of 24 Be prepared to explain your reasoning 4 Decide which of the following expressions would have the same value as 250 10 Be prepared to share your reasoning a 250 0 1 c 2 5 1 b 25 1 d 2 5 0 1 e 2 500 100 f 0 25 0 01 ACTIVITY 2 RECAP 43 180 What happens to the value of the quotient when both the divisor and the dividend are multiplied by the same power of 10 Use examples to show your thinking 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 12 Lesson Summary We know that fractions such as 6 4 and 60 40 are equivalent because Both the numerator and denominator of 60 40 have a factor of 10 so it can be written as Both fractions can be simplified to 600 divided by 400 is 1 5 and 60 divided by 40 is also 1 5 3 2 6 4 Just like fractions division expressions can be equivalent For example the expressions 540 90 and 5 400 900 are both equivalent to 54 9 because They all have a quotient of 6 The dividend and the divisor in 540 90 are each 10 times the dividend and divisor in 54 9 Those in 5 400 900 are each 100 times the dividend and divisor in 54 9 In both cases the quotient does not change This means that an expression such as 5 4 0 9 also has the same value as 54 9 Both the dividend and 1 divisor of 5 4 0 9 are 10 of those in 54 9 In general multiplying a dividend and a divisor by the same number does not change the quotient Multiplying by powers of 10 e g 10 100 1 000 etc can be particularly useful for dividing decimals as we will see in an upcoming lesson 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M5 LESSON 12 Name Date GRADE 6 MISSION 5 LESSON 12 Exit Ticket 1 Use long division to find the value of 43 5 3 If you get stuck you can draw base ten diagrams Be sure to say what each type of figure represents in your diagrams 2 Explain why all of these expressions have the same value 100 5 10 0 5 1 0 05 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 183

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ZEARN MATH STUDENT EDITION G6M5 LESSON 13 Lesson 13 Dividing Decimals by Decimals Let s divide decimals by decimals Warm Up 1 2 Use long division to find the value of 5 04 7 Which of the following quotients has the same value as 5 04 7 Be prepared to explain how you know a 5 04 70 b 50 4 70 c 504 000 700 d 504 000 700 000 Concept Exploration ACTIVITY 1 3 Think of one or more ways to find 3 0 12 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 185

G6M5 LESSON 13 43 ZEARN MATH STUDENT EDITION Solve the problems about dividing with decimals 1 Find 1 8 0 004 Show your reasoning If you get stuck think about what equivalent division expression you could write to help you divide 2 Diego said To divide decimals we can start by moving the decimal point in both the dividend and divisor by the same number of places and in the same direction Then we find the quotient of the resulting numbers Do you agree with Diego s statement Use the division expression 7 5 1 25 to support your answer ACTIVITY 2 53 186 Find each quotient using a method of your choice Then discuss your calculations with your group and agree on the correct answers 1 106 5 3 2 58 8 0 7 3 257 4 1 1 4 Mai is making friendship bracelets Each bracelet is made from 24 3 cm of string If she has 170 1 cm of string how many bracelets can she make Explain or show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M5 LESSON 13 Lesson Summary One way to find a quotient of two decimals is to multiply each decimal by a power of 10 so that both products are whole numbers If we multiply both decimals by the same power of 10 this does not change the value of the quotient For example the quotient 7 65 1 2 can be found by multiplying the two decimals by 10 or by 100 and instead finding 76 5 12 or 765 120 To calculate 765 120 which is equivalent to 76 5 12 we could use basessten diagrams partial quotients or long division Here is the calculation with long division 6 375 120 765 720 45 0 36 0 9 00 8 40 600 600 0 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 187

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ZEARN MATH STUDENT EDITION Name G6M5 LESSON 13 Date GRADE 6 MISSION 5 LESSON 13 Exit Ticket 1 Write two division expressions that have the same value as 36 8 2 3 2 Find the value of 36 8 2 3 Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 189

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ZEARN MATH STUDENT EDITION G6M5 LESSON 14 Lesson 14 Using Operations on Decimals to Solve Problems Let s solve some problems using decimals Warm Up 1 1 For each expression choose the best estimate of its value 76 2 15 0 5 5 50 2 56 34 48 1 10 100 3 124 3 20 6 60 600 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 14 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 192 There are 10 equally spaced hurdles on a race track The first hurdle is 13 72 meters from the start line The final hurdle is 14 02 meters from the finish line The race track is 110 meters long 1 Draw a diagram that shows the hurdles on the race track Label all known measurements 2 How far are the hurdles from one another Explain or show your reasoning 3 A professional runner takes 3 strides between each pair of hurdles The runner leaves the ground 2 2 meters before the hurdle and returns to the ground 1 meter after the hurdle About how long are each of the runner s strides between the hurdles Show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M5 LESSON 14 Lesson Summary Diagrams can help us communicate and model mathematics A clearly labeled diagram helps us visualize what is happening in a problem and accurately communicate the information we need Sports offer great examples of how diagrams can help us solve problems For example to show the placement of the running hurdles in a diagram we needed to know what the distances 13 72 and 14 02 meters tell us and the number of hurdles to draw An accurate diagram not only helped us set up and solve the problem correctly but also helped us see that there are only nine spaces between ten hurdles To communicate information clearly and solve problems correctly it is also important to be precise in our measurements and calculations especially when they involve decimals In tennis for example the length of the court is 23 77 meters Because the boundary lines on a tennis court have a significant width we would want to know whether this measurement is taken between the inside of the lines the center of the lines or the outside of the lines Diagrams can help us attend to this detail as shown here The accuracy of this measurement matters to the tennis players who use the court so it matters to those who paint the boundaries as well The tennis players practice their shots to be on or within certain lines If the tennis court on which they play is not precisely measured their shots may not land as intended in relation to the boundaries Court painters usually need to be sure their measurements are 1 accurate to within 100 of a meter or one centimeter outside 23 77 m center 23 77 m 23 77 23 77 m m inside 23 77 23 77 m 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 193

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ZEARN MATH STUDENT EDITION Name G6M5 LESSON 14 Date GRADE 6 MISSION 5 LESSON 14 Exit Ticket Andre is running in an 80 meter hurdle race There are 8 equally spaced hurdles on the race track The first hurdle is 12 meters from the start line and the last hurdle is 15 5 meters from the finish line 1 Estimate how far the hurdles are from one another Explain your reasoning 2 Calculate how far the hurdles are from one another 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 195

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ZEARN MATH STUDENT EDITION G6M5 LESSON 15 Lesson 15 Making and Measuring Boxes Let s use what we know about decimals to make and measure boxes Warm Up 1 1 2 3 Your teacher will demonstrate how to make an open top box by folding a sheet of paper Your group will receive 3 or more sheets of square paper Each person in your group will make 1 box Before you begin folding Record the side lengths of your papers from the smallest to the largest Paper for Box 1 cm Paper for Box 2 cm Paper for Box 3 cm Compare the side lengths of the square sheets of paper Be prepared to explain how you know a The side length of the paper for Box 2 is times the side length of the paper for Box 1 b The side length of the paper for Box 3 is times the side length of the paper for Box 1 Make some predictions about the measurements of the three boxes your group will make The surface area of Box 3 will be as large as that of Box 1 Box 2 will be times as tall as Box 1 Box 3 will be times as tall as Box 1 Now you are ready to fold your paper into a box 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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

G6M5 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Now that you have made your boxes you will measure them and check your predictions about how their heights and surface areas compare 1 a Measure the length and height of each box to the nearest tenth of a centimeter Record the measurements in the table Side length of paper cm Length of box cm Height of box cm Surface area sq cm Box 1 Box 2 Box 3 b Calculate the surface area of each box Show your reasoning and decide on an appropriate level of precision for describing the surface area Is it the nearest 10 square centimeters nearest square centimeter or something else Record your answers in the table 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 2 To see how many times as large one measurement is when compared to another we can compute their quotient Divide each measurement of Box 2 by the corresponding measurement for Box 1 to complete the following statements a The length of Box 2 is times the length of Box 1 b The height of Box 2 is times the height of Box 1 c 3 G6M5 LESSON 15 The surface area of Box 2 is times the surface area of Box 1 Find out how the dimensions of Box 3 compare to those of Box 1 by computing quotients of their lengths heights and surface areas Show your reasoning a The length of Box 3 is ________ times the length of Box 1 b The height of Box 3 is ________ times the height of Box 1 c The surface area of Box 3 is ________ times the surface area of Box 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 199

G6M5 LESSON 15 4 ZEARN MATH STUDENT EDITION Record your results in the table Side length of paper Length of box Height of box Surface area Box 2 compared to Box 1 Box 3 compared to Box 1 5 200 Earlier in the warm up you made predictions about how the heights and surface areas of the two larger boxes would compare to those of the smallest box Discuss with your group How accurate were your predictions Were they close to the results you found by performing calculations Let s say you had another piece of square paper to make Box 4 If the side length of this paper is 4 times the side length of the paper for Box 1 predict how the length height and surface area of Box 4 would compare to those of Box 1 How did you make your prediction 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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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G6M6 LESSON 1 ZEARN MATH STUDENT EDITION Lesson 1 Tape Diagrams and Equations Let s see how tape diagrams and equations can show relationships between amounts Warm Up 1 Below are two diagrams One represents 2 5 7 The other represents 5 2 10 Determine the equation that matches up with each diagram Then label the length of each diagram 2 2 1 2 2 2 2 2 5 Draw a diagram that represents each equation 4 3 7 2 4 3 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 203

G6M6 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 Here are two tape diagrams Match each equation to one of the tape diagrams x 4 x x 12 x 12 List the equations that match up with this tape diagram here 204 x List the equations that match up with this tape diagram here 1 4 x 12 4 12 4 x 7 12 4 x 2 12 4 x 5 12 x 4 8 x 12 4 3 4 x 12 6 12 4 x 9 x x x x 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 G6M6 LESSON 1 ACTIVITY 2 43 1 For each equation draw a diagram Then find the value of the unknown that makes the equation true 18 3 x Tape Diagram Solve for x 2 18 3 y Tape Diagram Solve for 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 205

G6M6 LESSON 1 ZEARN MATH STUDENT EDITION Lesson Summary Tape diagrams can help us understand relationships between quantities and how operations describe those relationships A B x x x 21 y 3 21 Diagram A has 3 parts that add to 21 Each part is labeled with the same letter so we know the three parts are equal Here are some equations that all represent diagram A x x x 21 3 x 21 x 21 3 x 1 3 21 Notice that the number 3 is not seen in the diagram the 3 comes from counting 3 boxes representing 3 equal parts in 21 We can use the diagram or any of the equations to reason that the value of x the value of x is 7 is 7 Diagram B has 2 parts that add to 21 Here are some equations that all represent diagram B y 3 21 y 21 3 We can use the diagram or any of the equations to reason that the value of y the value of y is 18 is 18 3 21 y 206 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 1 Date GRADE 6 MISSION 6 LESSON 1 Exit Ticket Finish the first diagram so that it represents 5 x 15 and the second diagram so that it represents 5 y 15 15 15 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 207

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ZEARN MATH STUDENT EDITION G6M6 LESSON 2 Lesson 2 Truth and Equations Let s use equations to represent stories and see what it means to solve equations Warm Up 1 1 The equation a b c could be true or false Answer the questions below The equation a b c could be true or false a If a is 3 b is 4 and c is 5 is the equation true or false b Find new values of a b and c that make the equation true c 2 1 Find new values of a b and c that make the equation false The equation x y z could be true or false Answer the questions below The equation x y z could be true or false a If x is 3 y is 4 and z is 12 is the equation true or false b Find new values of x y and z that make the equation true c Find new values of x y and z that make the equation false 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 209

G6M6 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 3 210 Below are three situations Which equation best represents each situation Consider drawing a diagram 1 After Elena ran 5 miles on Friday she had run a total of 20 miles for the week She ran x miles before Friday 2 Andre s school has 20 clubs which is five times as many as his cousin s school His cousin s school has x clubs 3 Jada volunteers at the animal shelter She divided 5 cups of cat food equally to feed 20 cats Each cat received x cups of food x 5 20 x 20 5 5x 20 x 20 5 5 20 x 20x 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

ZEARN MATH STUDENT EDITION G6M6 LESSON 2 ACTIVITY 2 43 Here are some equations that contain a variable and a list of values Think about what each equation means and find a solution in the list of values below Consider drawing a diagram Be prepared to explain why your solution is correct 1 8 1 12 6 b 4 1 3 7 2 8c 8 4 7 8 5 3 5 9 5 1 1000 a 400 2 3 4 2 3 d 10 9 Values 5 10e 1 5 3 16 7 6 10 0 5f 7 3 20 7 0 99 1 g 0 01 400 0 1 600 0 5 1400 8 h 3 7 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 211

G6M6 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary An equation can be true or false An example of a true equation is 7 1 4 2 An example of a false equation is 7 1 9 An equation can have a letter in it for example u 1 8 This equation is false if u is 3 because 3 1 does not equal 8 This equation is true if u is 7 because 7 1 8 A letter in an equation is called a variable In u 1 8 the variable is u A number that can be used in place of the variable that makes the equation true is called a solution to the equation In u 1 8 the solution is 7 When a number is written next to a variable the number and the variable are being multiplied For example 7x 21 means the same thing as 7 x 21 A number written next to a variable is called a coefficient If no coefficient is written the coefficient is 1 For example in the equation p 3 5 the coefficient of p is 1 TERMINOLOGY Coefficient Solution to an equation Variable 212 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 2 Date GRADE 6 MISSION 6 LESSON 2 Exit Ticket 1 Explain how you know 88 is a solution to the equation 8 x 11 by completing the sentences The word solution means 88 is a solution to 1 8 x 11 because 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 213

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ZEARN MATH STUDENT EDITION G6M6 LESSON 3 Lesson 3 Stay in Balance Let s use balanced hangers to help us solve equations Warm Up 1 1 For each diagram find one thing that must be true one thing that could be true and one thing that cannot possibly be true For diagram A find a One thing that must be true A b One thing that could be true or false c 2 One thing that cannot possibly be true For diagram B find a One thing that must be true B b One thing that could be true or false c One thing that cannot possibly be true 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 215

G6M6 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Match each hanger with an equation Then solve for the value of each variable A B 1 1 1 1 1 w 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Match each hanger to an equation Complete the equation by writing x y z or w in the empty box 3 6 2 D C 3 6 6 1 6 3 Find a solution to each equation Use the hangers to explain what each solution means ACTIVITY 2 3 1 Below are some balanced hangers Each piece is labeled with its weight Diagram A a Write an equation b Explain how to reason with the diagram to find the weight of the piece labeled x c 216 A 3 8 Explain how to reason with the equation to find the weight of the piece labeled x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION 2 Diagram B a Write an equation G6M6 LESSON 3 B 12 b Explain how to reason with the diagram to find the weight of the pieces labeled y c 3 Explain how to reason with the equation to find the weight of the pieces labeled y Diagram C C a Write an equation 11 b Explain how to reason with the diagram to find the weight of the pieces labeled z c 4 Explain how to reason with the equation to find the weight of the pieces z Diagram D D a Write an equation b Explain how to reason with the diagram to find the weight of the piece labeled w c w 4 13 5 4 35 Explain how to reason with the equation to find the weight of the piece labeled w 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 217

G6M6 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary A hanger stays balanced when the weights on both sides are equal We can change the weights and the hanger will stay balanced as long as both sides are changed in the same way For example adding 2 pounds to each side of a balanced hanger will keep it balanced Removing half of the weight from each side will also keep it balanced An equation can be compared to a balanced hanger We can change the equation but for a true equation to remain true the same thing must be done to both sides of the equal sign If we add or subtract the same number on each side or multiply or divide each side by the same number the new equation will still be true This way of thinking can help us find solutions to equations Instead of checking different values we can think about subtracting the same amount from each side or dividing each side by the same number A B y 11 218 11 5 Diagram A can be represented by the equation 3x 11 Diagram B can be represented with the equation 11 y 5 If we break the 11 into 3 equal parts each part will have the same weight as a block with an x If we remove a weight of 5 from each side of the hanger it will stay in balance Splitting each side of the hanger into 3 equal parts is the same as dividing each side of the equation by 3 Removing 5 from each side of the hanger is the same as subtracting 5 from each side of the equation 3x divided by 3 is x 11 5 is 6 11 divided by 3 is 11 3 y 5 5 is y 3x 11 is true then x 11 3 is true If 11 y 5 is true then 6 y is true The solution to 3x 11 is 11 3 The solution to 11 y 5 is 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 Name G6M6 LESSON 3 Date GRADE 6 MISSION 6 LESSON 3 Exit Ticket Here is a balanced hanger w w 25 w w 1 Write an equation representing this hanger 2 Find the weight of one circle Show or explain how you found 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 219

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ZEARN MATH STUDENT EDITION G6M6 LESSON 4 Lesson 4 Practice Solving Equations and Representing Situations with Equations Let s solve equations by doing the same to each side Warm Up 1 Find the value of each expression mentally 5 2 5 2 1 5 2 17 5 2 78 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 221

G6M6 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Solve the equations in one column Your partner will work on the other column Check in with your partner after you finish each row Your answers in each row should be the same If your answers aren t the same work together to find the error and correct it Column A Column B 18 2x 36 4x 17 x 9 13 x 5 8x 56 3x 21 21 1 4 x 6x 45 x 4 5 7 1 5 5 6 9 x x 3 5 6 8 x 55 3 7 x 33 6x 1 3 10x 10 x 9 14 88 x 17 05 3 34 x 1 222 1 3 8x 60 2 17 x 5 20 3 28 1 4 6 17 x 9 14 5 7 x 15 3 91 x 6 08 7 5 7 x 15 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 4 ACTIVITY 2 3 Circle all of the equations that describe each situation Then find each solution Consider drawing a diagram to help you solve 1 Clare has 8 fewer books than Mai If Mai has 26 books how many books does Clare have 26 x 8 x 26 8 x 8 26 26 8 x x 2 A coach formed teams of 8 from all the players in a soccer league There are 14 teams How many players are in the league y 14 8 y 8 14 1 8 y 14 y 14 8 y 3 Kiran scored 223 more points in a computer game than Tyler If Kiran scored 409 points how many points did Tyler score 223 409 z 409 223 z 409 223 z 409 223 z z 4 Mai ran 27 miles last week which was three times as far as Jada ran How far did Jada run 3 w 27 w 1 3 27 w 27 3 w 3 27 w 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 223

G6M6 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary Writing and solving equations can help us answer questions about situations Suppose a scientist has 13 68 liters of acid and needs 16 05 liters for an experiment How many more liters of acid does she need for the experiment We can represent this situation with the equation 13 68 x 16 05 When working with hangers we saw that the solution can be found by subtracting 13 68 from each side This gives us some new equations that also represent the situation x 16 05 13 68 x 2 37 Finding a solution in this way leads to a variable on one side of the equal sign and a number on the other We can easily read the solution in this case 2 37 from an equation with a letter on one side and a number on the other We often write solutions in this way Let s say a food pantry takes a 54 pound bag of rice and splits it into portions that each weigh pound How many portions can they make from this bag of a We can represent this situation with the equation 3 4 3 4 x 54 We can find the value of x by dividing each side by represent the same situation x 54 3 4 This gives us some new equations that 3 4 x 72 224 The solution is 72 portions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 4 Name Date GRADE 6 MISSION 6 LESSON 4 Exit Ticket 1 2 1 Write a story to match the equation x 2 2 Explain what x represents in your story 3 Solve the equation Explain or show your reasoning 10 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 225

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ZEARN MATH STUDENT EDITION G6M6 LESSON 5 Lesson 5 A New Way to Interpret a over b Let s investigate what a fraction means when the numerator and denominator are not whole numbers Warm Up 1 Solve each equation Be prepared to explain your reasoning 1 0 07 10m 2 10 1 t 7 2 Concept Exploration ACTIVITY 1 2 Solve each equation 1 35 7x 2 35 11x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 227

G6M6 LESSON 5 3 7x 7 7 4 0 3x 2 1 5 2 5 1 2 ZEARN MATH STUDENT EDITION x ACTIVITY 2 Take turns with your partner telling a story that might be represented by each equation Then for each equation choose one story state what quantity x describes and solve the equation Consider using a diagram to help you solve 3 1 2 228 0 7 x 12 1 4 x 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M6 LESSON 5 Lesson Summary In the past you learned that a fraction such as 4 5 can be thought of in a few ways 4 5 is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0 4 5 is the share that each person would have if 4 wholes were shared equally among 5 people This means that 45 is the result of dividing 4 by 5 We can extend this meaning of a fraction as a division to fractions whose numerators and denominators are not whole numbers For example we can represent 4 5 pounds of rice divided into portions that each weigh 1 5 pounds as 4 5 1 5 4 5 1 5 3 Fractions that involve non whole numbers can also be used when we solve equations Suppose a road under construction is 38 finished and the length of the completed part is long will the road be when completed We can write the equation 3 8 The completed road will be 3 x 5 9 4 3 4 3 miles How to represent the situation and solve the equation or about 3 6 miles long 3 8 x x 4 3 3 8 4 3 x 4 3 x 32 9 3 8 3 5 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 229

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ZEARN MATH STUDENT EDITION G6M6 LESSON 5 Name Date GRADE 6 MISSION 6 LESSON 5 Exit Ticket 2 Select all the expressions that are solutions to 5 3 x a 5 b 2 3 5 2 3 c 5 d 15 2 e 10 3 2 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 231

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ZEARN MATH STUDENT EDITION G6M6 LESSON 6 Lesson 6 Write Expressions Where Letters Stand for Numbers Let s use expressions with variables to describe situations Warm Up 1 If x is 6 what is x 4 7 x x2 1 3x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 233

G6M6 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Lin set up a lemonade stand She sells the lemonade for 0 50 per cup a Complete the table to show how much money she would collect if she sold each number of cups Lemonade sold number of cups 12 c 183 Money collected dollars b How many cups did she sell if she collected 127 50 Be prepared to explain your reasoning 3 Elena is 59 inches tall Some other people are taller than Elena a Complete the table to show the height of each person Person Andre How much taller than Elena inches 4 Lin 6 1 2 Noah d Person s height inches b If Noah is 64 234 3 4 inches tall how much taller is he than Elena 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 6 ACTIVITY 2 43 1 Solve the following problems Be prepared to explain your reasoning Clare is 5 years older than her cousin a How old would Clare be if her cousin is 10 years old 2 years old x years old b Clare is 12 years old How old is Clare s cousin 2 Diego has 3 times as many comic books as Han a How many comic books does Diego have if Han has 6 comic books n books b Diego has 27 comic books How many comic books does Han have 3 Two fifths of the vegetables in Priya s garden are tomatoes a How many tomatoes are there if Priya s garden has 20 vegetables x vegetables b Priya s garden has 6 tomatoes How many total vegetables are there 4 A school paid 31 25 for each calculator a If the school bought x calculators how much did they pay b The school spent 500 on calculators How many did the school buy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 235

G6M6 LESSON 6 ZEARN MATH STUDENT EDITION Lesson Summary Suppose you share a birthday with a neighbor but she is 3 years older than you When you were 1 she was 4 When you were 9 she was 12 When you are 42 she will be 45 If we let a represent your age at any time your neighbor s age can be expressed a 3 Your age 1 9 42 a Neighbor s age 4 12 45 a 3 We often use a letter such as x or a as a placeholder for a number in expressions These are called variables just like the letters we used in equations previously Variables make it possible to write expressions that represent a calculation even when we don t know all the numbers in the calculation How old will you be when your neighbor is 32 Since your neighbor s age is calculated with the expression a 3 we can write the equation a 3 32 When your neighbor is 32 you will be 29 because a 3 32 is true when a is 29 236 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G6M6 LESSON 6 Date GRADE 6 MISSION 6 LESSON 6 Exit Ticket A plant measured x inches tall last week and 8 inches tall this week 1 2 Circle the expression that represents the number of inches the plant grew this week Explain how you know x 8 8 x For the expression not chosen describe a situation that the expression might 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 237

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ZEARN MATH STUDENT EDITION G6M6 LESSON 7 Lesson 7 Revisit Percentages Let s use equations to find percentages Warm Up 1 Solve each problem mentally 1 Bottle A contains 4 ounces of water which is 25 of the amount of water in Bottle B How much water is there in Bottle B 2 Bottle C contains 150 of the water in Bottle B How much water is there in Bottle C 3 Bottle D contains 12 ounces of water What percentage of the amount of water in Bottle B is this 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 239

G6M6 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Answer the following questions in your notes Answer each question and show your reasoning a Is 60 of 400 equal to 87 b Is 60 of 200 equal to 87 c Is 60 of 120 equal to 87 2 60 of x is equal to 87 Write an equation that expresses the relationship between 60 x and 87 Solve your equation 3 Write an equation to help you find the value of each variable Solve the equation a 60 of c is 43 2 b 38 of e is 190 240 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M6 LESSON 7 ACTIVITY 2 3 Answer the following questions and be ready to explain your reasoning 1 Puppy A weighs 8 pounds which is about 25 of its adult weight What will be the adult weight of Puppy A 2 Puppy B weighs 8 pounds which is about 75 of its adult weight What will be the adult weight of Puppy B 3 If you haven t already write an equation for each situation Then show how you could find the adult weight of each puppy by solving the equation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 241

G6M6 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary Students have been solving equations with fraction coefficients in the past few lessons so these percent problems are an application of their prior work Consider asking some of the following questions to guide the discussion and help students recognize this connection 242 How are the equations we wrote today related to the equations we have previously written with fractions How do solution strategies compare Can equations be used to solve other types of problems with percents For example where we know the part and the whole but not what percent the part is of the whole Yes For 25 example the equation 20p 5 and its solution p 14 or 100 tells us that 5 is 25 of 20 Describe a situation where you know what percent a number is of another but you don t know that second number Explain to a partner how you would find the second number 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION Name G6M6 LESSON 7 Date GRADE 6 MISSION 6 LESSON 7 Exit Ticket Noah raised 54 to support the animal shelter which is 60 of his fundraising goal 1 Write an equation to represent the situation 2 What is Noah s fundraising goal Show or explain how you found 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 243

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ZEARN MATH STUDENT EDITION G6M6 LESSON 8 Lesson 8 Equal and Equivalent Let s use diagrams to figure out which expressions are equivalent and which are just sometimes equal Warm Up 1 Find a solution to each equation mentally 3 x 8 10 12 x x2 49 1 3x 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 245

G6M6 LESSON 8 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Draw diagrams that show 2 3 3 2 2 3 does not equal 2 3 3 Answer the following questions Here is a diagram of x 2 and 3x when x is 4 Notice that the two diagrams are lined up on their left sides x x 246 2 x x 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M6 LESSON 8 In each of your drawings below line up the diagrams on one side 1 Draw a diagram of x 2 and a separate diagram of 3x when x is 3 2 Draw a diagram of x 2 and a separate diagram of 3x when x is 2 3 Draw a diagram of x 2 and a separate diagram of 3x when x is 1 4 Draw a diagram of x 2 and a separate diagram of 3x when x is 0 5 When are x 2 and 3x equal When are they not equal Use your diagrams to explain 6 Draw a diagram of x 3 and a separate diagram of 3 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 247

G6M6 LESSON 8 7 ZEARN MATH STUDENT EDITION When are x 3 and 3 x equal When are they not equal Use your diagrams to explain ACTIVITY 2 Here is a list of expressions Find any pairs of expressions that are equivalent If you get stuck try reasoning with diagrams 43 a 3 a a a a 13 a 3 1 3a 3a a 3 1a a 3 a 248 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 8 Lesson Summary We can use diagrams showing lengths of rectangles to see when expressions are equal For example the expressions x 9 and 4x are equal when x is 3 but are not equal for other values of x x x 9 when x 1 x x x x 4x when x 1 x x 9 when x 2 x x x x 4x when x 2 x x x 9 when x 3 x x x 4x when x 3 x x x 9 when x 4 x x x 4x when x 4 Sometimes two expressions are equal for only one particular value of their variable Other times they seem to be equal no matter what the value of the variable Expressions that are always equal for the same value of their variable are called equivalent expressions However it would be impossible to test every possible value of the variable How can we know for sure that expressions are equivalent We use the meaning of operations and properties of operations to know that expressions are equivalent Here are some examples x 3 is equivalent to 3 x because of the commutative property of addition 4 y is equivalent to y 4 because of the commutative property of multiplication a a a a a is equivalent to 5 a because adding 5 copies of something is the same as multiplying it by 5 b 3 is equivalent to b 13 because dividing by a number is the same as multiplying by its reciprocal In the coming lessons we will see how another property the distributive property can show that expressions are equivalent TERMINOLOGY Equivalent expressions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 249

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ZEARN MATH STUDENT EDITION Name G6M6 LESSON 8 Date GRADE 6 MISSION 6 LESSON 8 Exit Ticket Decide if the expressions in each pair are equivalent Explain how you know 1 x x x x and 4x 2 5x and x 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 251

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ZEARN MATH STUDENT EDITION G6M6 LESSON 9 Lesson 9 The Distributive Property Part 1 Let s use the distributive property to make calculating easier Warm Up 1 Find each product mentally 1 1 5 102 2 2 5 98 3 3 5 999 Concept Exploration ACTIVITY 1 2 Answer the questions about the two diagrams B A 4 6 3 1 2 7 2 Select all the expressions that represent the area of the large outer rectangle in figure A Explain your reasoning a 6 3 2 b 6 3 6 2 c 6 3 2 d 6 5 e 6 3 2 f 6 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 253

G6M6 LESSON 9 2 ZEARN MATH STUDENT EDITION Select all the expressions that represent the area of the shaded rectangle on the left side of figure B Explain your reasoning a 4 7 4 2 b 4 7 2 c 4 5 d 4 7 4 2 e 4 7 2 f 4 7 2 g 4 2 4 7 ACTIVITY 2 3 Complete the table If you get stuck skip an entry and come back to it or consider drawing a diagram of two rectangles that share a side Column 1 Column 2 Column 3 Column 4 Value 5 98 5 100 2 5 100 5 2 500 10 490 33 12 33 10 2 3 10 3 4 30 12 100 0 04 0 06 8 1 2 8 1 4 9 12 24 16 254 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 9 Lesson Summary When we need to do mental calculations we often come up with ways to make the calculation easier to do mentally Suppose we are grocery shopping and need to know how much it will cost to buy 5 cans of beans at 79 cents a can We may calculate mentally in this way 5 79 5 70 5 9 350 45 395 In general when we multiply two numbers or factors we can break up one of the factors into parts multiply each part by the other factor and then add the products The result will be the same as the product of the two original factors When we break up one of the factors and multiply the parts we are using the distributive property The distributive property also works with subtraction Here is another way to find 5 79 5 79 5 80 1 400 5 395 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 255

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ZEARN MATH STUDENT EDITION Name G6M6 LESSON 9 Date GRADE 6 MISSION 6 LESSON 9 Exit Ticket Write a number or expression in each empty space to create true equations 1 7 3 5 2 15 10 3 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 257

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ZEARN MATH STUDENT EDITION G6M6 LESSON 10 Lesson 10 The Distributive Property Part 2 Let s use rectangles to understand the distributive property with variables Warm Up 1 Solve the problems about rectangles and their areas 1 A rectangle has a width of 4 units and a length of m units Write an expression for the area of this rectangle 2 What is the area of the rectangle if m is 3 units 2 2 units 3 Could the area of this rectangle be 11 square units Why or why not 1 5 unit Concept Exploration ACTIVITY 1 2 1 Solve the problems about rectangles and their areas Here are two rectangles The length and width of one rectangle are 8 and 5 The width of the other rectangle is 5 but its length is unknown so we labeled it x Write an expression for the sum of the areas of the two rectangles x 8 5 2 5 The two rectangles can be composed into one larger rectangle as shown What are the width and length of the new large rectangle x 5 3 8 5 Write an expression for the total area of the large rectangle as the product of its width and its length 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 259

G6M6 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 For each rectangle write expressions for the length and width and two expressions for the total area Record them in the table Check your expressions in each row with your group and discuss any disagreements 3 A a B 5 1 3 3 D p p p E p 6 Length C 1 1 1 r 6 8 F 3x 8 5 m Width 260 x 6 Area as a product of length times width Area as a sum of the areas of the smaller rectangles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 10 Lesson Summary Here is a rectangle composed of two smaller rectangles A and B 3 2 x A B Based on the drawing we can make several observations about the area of the rectangle One side length of the large rectangle is 3 and the other is 2 x so its area is 3 2 x Since the large rectangle can be decomposed into two smaller rectangles A and B with no overlap the area of the large rectangle is also the sum of the areas of rectangles A and B 3 2 3 x or 6 3x Since both expressions represent the area of the large rectangle they are equivalent to each other 3 2 x is equivalent to 6 3x We can see that multiplying 3 by the sum 2 x is equivalent to multiplying 3 by 2 and then 3 by x and adding the two products This relationship is an example of the distributive property 3 2 x 3 2 3 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 261

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ZEARN MATH STUDENT EDITION G6M6 LESSON 10 Name Date GRADE 6 MISSION 6 LESSON 10 Exit Ticket Select all the expressions that represent the large rectangle s total area b 3 5 1 3 5 b 2 5 b 3 3 5b 15 4 15 5b 5 3 5 3b 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 263

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ZEARN MATH STUDENT EDITION G6M6 LESSON 11 Lesson 11 The Distributive Property Part 3 Let s practice writing equivalent expressions by using the distributive property Warm Up A rectangle with dimensions 6 cm and w cm is partitioned into two smaller rectangles 1 Explain why each of these expressions represents the area in cm2 of the shaded portion 4 6 6w 24 w 6 w 4 Concept Exploration ACTIVITY 1 Match each expression in Column 1 to an equivalent expression in Column 2 If you get stuck consider drawing a diagram 2 Column 1 Column 2 a a 1 2 3 1 3 4a b b 2 12 4 2 12 2 4 2 c 3 2 3a 5b 4 2 3 a e 6a 10b 5 a 2a 3a f 6 10a 12 7 2 a d 12a 3b 2 3 15a 18 0 4 5 2 5a g 2a 3a 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 265

G6M6 LESSON 11 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 The distributive property can be used to write equivalent expressions In each row use the distributive property to write an equivalent expression If you get stuck draw a diagram Product Sum or difference 3 3 x 4x 20 9 5 x 4x 7x 3 2x 1 10x 5 x 2x 3x 1 2 x 6 y 3x 4z 2xyz 3yz 4xz Lesson Summary The distributive property can be used to write a sum as a product or write a product as a sum You can always draw a partitioned rectangle to help reason about it but with enough practice you should be able to apply the distributive property without making a drawing Here are some examples of expressions that are equivalent due to the distributive property 9 18 9 1 2 2 3x 4 6x 8 2n 3n n n 2 3 1 11b 99a 11 b 9a k c d e kc kd ke 266 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 11 Date GRADE 6 MISSION 6 LESSON 11 Exit Ticket 1 Use the distributive property to write an expression that is equivalent to 12 4x 2 Draw a diagram that shows the two expressions are equivalent 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 267

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ZEARN MATH STUDENT EDITION G6M6 LESSON 12 Lesson 12 Meaning of Exponents Let s see how exponents show repeated multiplication Warm Up 1 What do you notice What do you wonder 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 269

G6M6 LESSON 12 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 You are walking along and you find a brass bottle that looks really really old There appears to be some writing on the bottle You try to clean off some dirt to read it better A genie appears He is so happy to be free He wants to repay you He offers two ways to repay you and you must choose one He will give you 50 000 or He will give you one magical 1 coin The magic coin will turn into 2 coins on the first day The 2 coins will turn into 4 coins on the second day On the third day the 4 coins will magically turn into 8 coins The genie explains that the magic coins will continue this pattern for 28 days 270 1 The number of coins on the third day will be 2 2 2 Can you write another expression using exponents for the number of coins there will be on the third day 2 What do 25 and 26 represent in this situation Evaluate 25 and 26 without a calculator 3 How many days would it take for the number of magical coins to exceed 50 000 4 Will the value of the magical coins exceed a million dollars within the 28 days Explain or show your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M6 LESSON 12 ACTIVITY 2 3 Here are some expressions All but one of them equals 16 Find the one that is not equal to 16 and explain how you know 23 2 43 42 25 2 82 Write three expressions containing exponents so that each expression equals 81 Lesson Summary When we write an expression like 2n we call n the exponent If n is a positive whole number it tells how many factors of 2 we should multiply to find the value of the expression For example 21 2 and 25 2 2 2 2 2 There are different ways to say 25 We can say two raised to the power of five or two to the fifth power or just two to the fifth 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 271

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ZEARN MATH STUDENT EDITION Name G6M6 LESSON 12 Date GRADE 6 MISSION 6 LESSON 12 Exit Ticket 35 equals 243 Explain how to use that fact to quickly evaluate 36 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 273

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ZEARN MATH STUDENT EDITION G6M6 LESSON 13 Lesson 13 Expressions with Exponents Let s use the meaning of exponents to decide if equations are true Warm Up Which one doesn t belong 1 2 2 2 2 24 16 4 2 Concept Exploration ACTIVITY 1 Decide whether each equation is true or false and explain how you know 2 1 24 2 4 2 3 3 3 3 3 35 3 53 5 5 5 4 23 32 5 161 82 1 2 7 12 4 1 8 8 82 43 6 1 2 1 2 1 2 4 1 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 275

G6M6 LESSON 13 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 In each list find expressions that are equivalent to each other and explain to your partner why they are equivalent Your partner listens to your explanation If you disagree explain your reasoning until you agree Switch roles for each list There may be more than two equivalent expressions in each list 1 2 a 5 5 a 43 b 25 b 34 c c 52 d 2 5 d 4 4 4 3 4 a 6 6 6 a 115 b 63 b 11 11 11 11 11 c c 36 d 3 6 11 5 d 511 5 6 a b c d 276 4 4 4 1 1 5 5 15 3 1 15 1 125 1 5 a 53 2 b 35 2 c d 10 6 25 9 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

ZEARN MATH STUDENT EDITION G6M6 LESSON 13 Lesson Summary When working with exponents the bases don t have to always be whole numbers They can also be other kinds of numbers like fractions decimals and even variables For example we can use exponents in each of the following ways 23 4 2 3 2 3 2 3 2 3 1 7 3 1 7 1 7 1 7 x5 x x x x 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 277

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ZEARN MATH STUDENT EDITION Name G6M6 LESSON 13 Date GRADE 6 MISSION 6 LESSON 13 Exit Ticket Andre and Elena knew that after 28 days they would have 228 coins but they wanted to find out how many coins that actually is Andre wrote 228 2 28 56 Elena said No exponents mean repeated multiplication It should be 28 28 which works out to be 784 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 279

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ZEARN MATH STUDENT EDITION G6M6 LESSON 14 Lesson 14 Evaluating Expressions with Exponents Let s find the values of expressions with exponents Warm Up 1 Based on the given information what other measurements of the square and cube could we find 3 3 3 3 3 Concept Exploration ACTIVITY 1 2 A cube has a side length of 10 inches Jada says the surface area of the cube is 600 in2 and Noah says the surface area of the cube is 3 600 in2 Here is how each of them reasoned Jada s Method Noah s Method 6 102 6 102 6 100 602 600 3 600 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 281

G6M6 LESSON 14 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 52 4 22 25 24 5 23 10 3 42 12 22 20 23 1 33 9 21 3 61 1 9 282 Evaluate the expressions in one of the columns Your partner will work on the other column Check with your partner after you finish each row Your answers in each row should be the same If your answers aren t the same work together to find the error 12 3 1 8 13 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

ZEARN MATH STUDENT EDITION G6M6 LESSON 14 Lesson Summary Exponents give us a new way to describe operations with numbers so we need to understand how exponents get along with the other operations we know When we write 6 42 we want to make sure everyone agrees about how to evaluate this Otherwise some people might multiply first and others compute the exponent first and different people would get different values for the same expression Earlier we saw situations in which 6 42 represented the surface area of a cube with side length 4 units When computing the surface area we evaluate 42 first or find the area of one face of the cube first and then multiply the result by 6 In many other expressions that use exponents the part with an exponent is intended to be evaluated first To make everyone agree about the value of expressions like 6 42 the convention is to evaluate the part of the expression with the exponent first Here are a couple of examples 6 42 45 52 6 16 45 25 96 70 If we want to communicate that 6 and 4 should be multiplied first and then squared then we can use parentheses to group parts together 6 4 2 242 576 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 283

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ZEARN MATH STUDENT EDITION G6M6 LESSON 14 Name Date GRADE 6 MISSION 6 LESSON 14 Exit Ticket Jada and Noah wanted to find the total volume of a cube and a rectangular prism They know the prism s volume is 20 cubic units and they know the cube has side lengths of 10 units Jada says the total volume is 27 000 cubic units Noah says it is 1 020 cubic units Here is how each of them reasoned Jada s Method Noah s Method 20 103 20 103 303 20 1 000 27 000 1 020 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 285

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ZEARN MATH STUDENT EDITION G6M6 LESSON 15 Lesson 15 Equivalent Exponential Expressions Let s investigate expressions with variables and exponents Warm Up 1 1 Answer the questions about exponents Find the values of 3x and 13 x for different values of x x 3x 13 x 1 2 3 4 2 What patterns do you notice Concept Exploration ACTIVITY 1 2 Evaluate each expression for the given value of x 1 3x2 when x is 10 2 3x2 when x is 3 x3 4 when x is 4 1 9 4 x3 4 when x is 1 2 5 9 x7 when x is 1 6 9 x7 when x is 1 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 287

G6M6 LESSON 15 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Find a solution to each equation in the list that follows Numbers in the list may be a solution to more than one equation and not all numbers in the list will be used 1 64 x2 2 64 x3 3 2x 32 4 x 25 3 5 16 9 6 2 25 2x 7 2x 24 8 43 8x x2 8 6 5 125 15 8 288 89 1 4 3 2 3 4 5 6 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 15 Lesson Summary In this lesson we saw expressions that used the letter x as a variable We evaluated these expressions for different values of x To evaluate the expression 2x3 when x is 5 we replace the letter x with 5 to get 2 53 This is equal to 2 125 or just 250 So the value of 2x3 is 250 when x is 5 To evaluate x8 when x is 4 we replace the letter x with 4 to get has a value of 2 when x is 4 2 42 8 16 8 which equals 2 So x2 8 We also saw equations with the variable x and had to decide what value of x would make the equation true Suppose we have an equation 10 3 90 and a list of possible solutions 1 2 3 9 11 The only value of x that makes the equation true is 2 because 10 32 10 3 3 which equals 90 So 2 is the solution to the equation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 289

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ZEARN MATH STUDENT EDITION G6M6 LESSON 15 Name Date GRADE 6 MISSION 6 LESSON 15 Exit Ticket Match each equation to a solution 1 2x 64 2 x 25 3 3 3 34 3x 4 16 25 x2 8 125 4 5 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 291

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ZEARN MATH STUDENT EDITION G6M6 LESSON 16 Lesson 16 Two Related Quantities Part 1 Let s use equations and graphs to describe relationships with ratios Warm Up 1 Which one would you choose Be prepared to explain your reasoning A 5 pound jug of honey for 15 35 Three 1 5 pound jars of honey for 13 05 15 35 5 lbs 1 5 lbs 1 5 lbs 1 5 lbs 5 3 for 13 0 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 293

G6M6 LESSON 16 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Lin needs to mix a specific color of paint for the set of the school play The color is a shade of orange that uses 3 parts yellow for every 2 parts red Complete the table to show different combinations of red and yellow paint that will make the shade of orange Lin needs Cups of red paint r Cups of yellow paint y 2 3 Total cups of paint t 6 20 18 14 16 50 42 294 2 Lin notices that the number of cups of red paint is always 25 of the total number of cups She writes the equation r 25 t to describe the relationship Which is the independent variable Which is the dependent variable Explain how you know 3 Write an equation that describes the relationship between r and y where is y the independent variable 4 Write an equation that describes the relationship between r and y where r is the independent variable 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 5 G6M6 LESSON 16 Use the points in the table to create two graphs that show the relationship between r and y Match each relationship to one of the equations you wrote y 48 44 40 36 cups of red paint cups of yellow paint y 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 32 28 24 20 16 12 8 4 4 8 12 16 20 24 28 32 36 40 44 48 cups of red paint x 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 x cups of yellow paint Lesson Summary Equations are very useful for describing sets of equivalent ratios Here is an example A pie recipe calls for 3 green apples for every 5 red apples We can create a table to show some equivalent ratios Green apples g Red apples r 3 5 6 10 9 15 12 20 We can see from the table that r is always 53 as large as g and that g is always write equations to describe the relationship between g and r 3 5 as large as r We can When we know the number of green apples and want to find the number of red apples we can write r 5 3 g In this equation if g changes r is affected by the change so we refer to g as the independent variable and r as the dependent variable We can use this equation with any value of g to find r If 270 green apples are used then red apples are used 5 3 270 or 450 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 295

When we know the number of red apples and want to find the number of green apples we can write g 35 r In this equation if r changes g is affected by the change so we refer to r as the independent variable and g as the dependent variable We can use this equation with any value of r to find g If 275 red apples are used then green apples are used 3 5 275 or 165 We can also graph the two equations we wrote to get a visual picture of the relationship between the two quantities TERMINOLOGY Dependent variable Independent variable 296

ZEARN MATH STUDENT EDITION G6M6 LESSON 16 Name Date GRADE 6 MISSION 6 LESSON 16 Exit Ticket A brownie recipe calls for 1 cup of sugar and 12 cup of flour to make one batch of brownies To make multiple batches the equation f 12 s where f is the number of cups of flour and s is the number of cups of sugar represents the relationship Which graph also represents the relationship Explain how you know C B 3 2 1 1 2 3 4 cups of sugar 5 4 cups of flour 4 cups of flour cups of flour A 3 2 1 1 2 3 4 cups of sugar 5 4 3 2 1 1 2 3 4 cups of sugar 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 297

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ZEARN MATH STUDENT EDITION G6M6 LESSON 17 Lesson 17 Two Related Quantities Part 2 Let s use equations and graphs to describe stories with constant speed Warm Up 1 Lin and Jada each walk at a steady rate from school to the library Lin can walk 13 miles in 5 hours and Jada can walk 25 miles in 10 hours They each leave school at 3 00 and walk 3 14 miles to the library What time do they each arrive Concept Exploration ACTIVITY 1 2 1 Diego Elena and Andre participated in a walk a thon to raise money for cancer research They each walked at a constant rate but their rates were different Complete the table to show how far each participant walked during the walk a thon Time in hours Miles walked by Diego Miles walked by Elena Miles walked by Andre 1 2 6 12 5 2 11 17 5 How fast was each participant walking in miles per hour 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 299

G6M6 LESSON 17 ZEARN MATH STUDENT EDITION 3 How long did it take each participant to walk one mile 4 Graph the progress of each person in the coordinate plane Use a different color for each participant 20 18 16 Distance walked miles 14 12 10 8 6 4 2 1 5 3 2 4 Time hours 5 Diego says that d 3t represents his walk where d is the distance walked in miles and t is the time in hours a Explain why d 3t relates the distance Diego walked to the time it took b Write two equations that relate distance and time one for Elena and one for Andre 300 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 17 6 Use the equations you wrote to predict how far each participant would walk at their same rate in 8 hours 7 For Diego s equation and the equations you wrote which is the dependent variable and which is the independent variable Lesson Summary Equations are very useful for solving problems with constant speeds Here is an example A boat is traveling at a constant speed of 25 miles per hour 1 How far can the boat travel in 3 25 hours 2 How long does it take for the boat to travel 60 miles We can write equations to help us answer questions like these Let s use t to represent the time in hours and d to represent the distance in miles that the boat travels When we know the time and want to find the distance we can write d 25t In this equation if t changes d is affected by the change so t is the independent variable and d is the dependent variable This equation can help us find d when we have any value of t In 3 25 hours the boat can travel 25 3 25 or 81 25 miles d When we know the distance and want to find the time we can write t 25 In this equation if d changes t is affected by the change so d is the independent variable and t is the dependent variable This equation can help us find t when for any value of d To travel 60 miles it will take 60 25 or 2 2 5 hours These problems can also be solved using important ratio techniques such as a table of equivalent ratios The equations are particularly valuable in this case because the answers are not round numbers or easy to quickly evaluate 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 301

G6M6 LESSON 17 ZEARN MATH STUDENT EDITION We can also graph the two equations we wrote to get a visual picture of the relationship between the two quantities 302 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 G6M6 LESSON 17 Date GRADE 6 MISSION 6 LESSON 17 Exit Ticket During a walk a thon Noah s time in hours t and distance in miles d are related by the equation 13 d t A graph of the equation includes the point 12 4 1 Identify the independent variable 2 What does the point 12 4 represent in this situation 3 What point would represent the time it took to walk 7 1 2 miles 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 303

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ZEARN MATH STUDENT EDITION G6M6 LESSON 18 Lesson 18 More Relationships Let s use graphs and equations to show relationships involving area volume and exponents Warm Up 1 Which one doesn t belong A C B D Lesson ACTIVITY 1 2 Mai is creating a rectangular banner to advertise the school play The material for the banner is sold by the square foot Mai has enough money to buy 36 square feet of material She is trying to decide on the length and width of the banner 1 If the length is 6 feet what is the width 2 If the length is 4 feet what is the width 3 If the length is 9 feet what is the width 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 305

G6M6 LESSON 18 ZEARN MATH STUDENT EDITION To find different combinations of length and width that give an area of 36 square feet Mai uses the equation w 36l where w is the width and l is the length Compare your strategy and Mai s method for finding the width How were they alike or different 5 Use several combinations of length and width to create a graph that shows the relationship between the side lengths of various rectangles with area 36 square feet width 4 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 length 306 6 Explain how the graph describes the relationship between length and width for different rectangles with area 36 7 Suppose Mai used the equation l 36 w to find the length for different values of the width Would the graph be different if she graphed length on the vertical axis and width on the horizontal axis 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 G6M6 LESSON 18 ACTIVITY 2 3 1 A cereal manufacturer needs to design a cereal box that has a volume of 225 cubic inches and a height that is no more than 15 inches The designers know that the volume of a rectangular prism can be calculated by multiplying the area of its base and its height Complete the table with pairs of values that will make the volume 225 in3 Height in 5 Area of base in2 9 712 12 75 15 Describe how you found the missing values for the table 3 Write an equation that shows how the area of the base A is affected by changes in the height h for different rectangular prisms with volume 225 in3 4 Plot the ordered pairs from the table on the graph to show the relationship between the area of the base and the height for different boxes with volume 225 in3 area of base 2 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 height 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 307

G6M6 LESSON 18 ZEARN MATH STUDENT EDITION ACTIVITY 3 43 A researcher who is studying mosquito populations collects the following data Day in the study d Number of mosquitoes n 1 2 3 4 5 2 4 8 16 32 1 The researcher said that for these five days the number of mosquitoes n can be found with the equation n 2 where d is the day in the study Explain why this equation matches the data 2 Use the ordered pairs in the table to graph the relationship between number of mosquitoes and day in the study for these five days Describe the graph Compare how the data equation and graph illustrate the relationship between the day in the study and the number of mosquitoes 35 number of mosquitoes 3 40 30 25 20 15 10 5 1 2 3 4 5 6 day of study 4 308 If the pattern continues how many mosquitoes will there be on day 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 G6M6 LESSON 18 Lesson Summary Equations can represent relationships between geometric quantities For instance If s is the side length of a square then the area A is related to A s2 Sometimes the relationships are more specific For example the perimeter P of a rectangle with length l and width w is P 2l 2w If we consider only rectangles with a length of 10 then the relationship between the perimeter and the width is P 20 2w Here is another example of an equation with an exponent expressing the relationship between quantities A super ball is dropped from 10 feet On each successive bounce it only goes 12 as high as on the previous bounce This means that on the first bounce the ball will bounce 5 feet high and then on the second bounce it will only go 212 feet high and so on We can represent this situation with an equation to find how high the super ball will bounce after any number of bounces To find how high the super ball bounces on the nth bounce we have to multiply 10 feet the initial height by 12 and multiply by 12 again for each bounce thereafter we need to do this n times So the height h of the ball on the nth bounce will be h 10 12 n In this equation the dependent variable h is affected by changes in the independent variable n Equations and graphs can give us insight into different kinds of relationships between quantities and help us answer questions and solve problems 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 309

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ZEARN MATH STUDENT EDITION Name G6M6 LESSON 18 Date GRADE 6 MISSION 6 LESSON 18 Exit Ticket The equation 14 P s relates the perimeter P of any square and its side length s A graph of the equation includes the point 12 3 1 What does the point 12 3 represent in this situation 2 What point would represent a square with perimeter 20 21 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 311

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ZEARN MATH STUDENT EDITION G6V2 Terminology Coefficient A coefficient is a number that is multiplied by a variable For example in the expression 3x 5 the coefficient of x is 3 In the expression y 5 the coefficient of y is 1 because y 1 y d 250 225 Dependent variable For example a boat travels at a constant speed of 25 miles per hour The equation d 25t describes the relationship between the boat s distance and time The dependent variable is the distance traveled because d is the result of multiplying 25 by t Distance traveled miles The dependent variable is the result of a calculation 200 175 150 125 100 75 50 25 2 4 6 8 10 t Time hours Equivalent expressions Equivalent expressions are always equal to each other If the expressions have variables they are equal whenever the same value is used for the variable in each expression For example 3x 4x is equivalent to 5x 2x No matter what value we use for x these expressions are always equal When x 3 both expressions equal 21 When x 10 both expressions equal 70 d 250 225 The independent variable is used to calculate the value of another variable For example a boat travels at a constant speed of 25 miles per hour The equation d 25t describes the relationship between the boat s distance and time The independent variable is time because t is multiplied by 25 to get d 200 Distance traveled miles Independent variable 175 150 125 100 75 50 25 Long division Long division is a way to show the steps for dividing numbers in decimal form It finds the quotient one digit at a time from left to right For example here is the long division for 57 4 2 4 6 8 10 t Time hours 14 25 4 57 00 4 17 16 10 8 20 20 0 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 313

G6V2 ZEARN MATH STUDENT EDITION Solution to an equation A solution to an equation is a number that can be used in place of the variable to make the equation true For example 7 is the solution to the equation m 1 8 because it is true that 7 1 8 The solution to m 1 8 is not 9 because 9 1 8 Variable A variable is a letter that represents a number You can choose different numbers for the value of the variable For example in the expression 10 x the variable is x If the value of x is 3 then 10 x 7 because 10 3 7 If the value of x is 6 then 10 x 4 because 10 6 4 314 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up 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 org NAME Grade 6 Student Edition Vol 1 Mission 1 Area and Surface Area Mission 2 Introducing Ratios Mission 3 Unit Rates and Percentages Mission 4 Dividing Fractions Mission 5 Arithmetic in Base Ten Mission 6 Expressions and Equations Student Edition Vol 2 Vol 3 Mission 7 Rational Numbers Mission 8 Data Sets and Distributions Mission 9 Putting It All Together G6 Vol 2 Zearnmath_SE_Grade6_Vol2 indd 1 Grade 6 Volume 2 MISSIONS 1 2 3 4 5 6 7 8 9 12 15 22 11 16 AM