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22. One container holds 1 78 quarts of water and a second container holds5 34 quarts of water. How many more quarts of water does the secondcontainer hold than the first container?23. Each of 15 students will give a 1 12 -minute speech in English class.a. How long will it take to give the speeches?b. If the teacher begins recording on a digital camera with an houravailable, is there enough time to record everyone if she gives a15-minute introduction at the beginning of class and every studenttakes a minute to get ready? Explain.c. How much time is left on the digital camera?FOCUS ON HIGHER ORDER THINKING24. Represent Real-World Problems Kate wants to buy a new bicycle froma sporting goods store. The bicycle she wants normally sells for 360. Thestore has a sale where all bicycles cost 56 of the regular price. What is thesale price of the bicycle?26. Justify Reasoning To multiply a whole number by a fraction, you canfirst write the whole number as a fraction by placing the whole numberin the numerator and 1 in the denominator. Does following this stepchange the product? Explain.84Unit 2Work Area Houghton Mifflin Harcourt Publishing Company25. Error Analysis To find the product 37 49 , Cameron simplified 37 to 17 and4then multiplied the fractions 17 and 49 to find the product63. What isCameron’s error?

Going Further4.1COMMONCOREInterpret and computequotients of fractions, andsolve word problems involvingdivision of fractions byfractions, e.g., by using visualfraction models and equationsto represent the problem.Transforming Equations?ESSENTIAL QUESTIONEXPLORE ACTIVITY6.NS.1How can you transform a division equation intoa multiplication equation?COMMONCORE6.NS.1Dividing FractionsDivision and multiplication are opposite, or inverse, operations. A division equation canbe rewritten as a multiplication equation using the concept of fact families.32Rewrite 32 as a multiplication equation.55A What is the dividend?This will be the product, or answer, in themultiplication equation. Write the product in the last set of boxes. Houghton Mifflin Harcourt Publishing CompanyB What is the quotient, or answer, in the division equation?This willbecome one of the two factors in the multiplication equation. Write thequotient in the first set of boxes.C What is the divisor?This will become the other factor in themultiplication equation. Write the divisor in the second set of boxes.Reflect1.Analyze How does the multiplication equation compare to the correspondingdivision equation?Going Further 4.184A

Rewriting Division as MultiplicationYou can rewrite a division expression as a multiplication expression bychanging the order of the terms.EXAMPLECOMMONCORE6.NS.157Rewrite 57 as a multiplication problem.88STEP 1The dividend becomes the product, or answer,in the multiplication equation.STEP 2The divisor becomes one of the factors in themultiplication equation.STEP 3The quotient becomes the other factor in themultiplication equation.575—4 —5—788575—3 —5—788PracticeComplete the table below by using the completed equation to fill in themissing fraction in the incomplete equation.1.7 6 74 12 27 6 2 122.1 3 13 91 133.4 9 18 2 94 9 288 99 569 89 175699 105910 955107 4 1 321 43 7144.5.6.84BMultiplicationUnit 2 Houghton Mifflin Harcourt Publishing CompanyDivision

4.2Getting ReadyCOMMONCOREModeling Fraction Division?6.NS.1Interpret and computequotients of fractions, andsolve word problems involvingdivision of fractions byfractions, e.g., by using visualfraction models and equationsto represent the problem.ESSENTIAL QUESTIONHow can you model fraction division?EXPLORE ACTIVITYCOMMONCORE6.NS.1Modeling DivisionJust like division of whole numbers, one method of solving a division problemwith fractions is to make a model.Model the division expression and find the quotient.15 3A To model 15, draw 15 dots.B To model dividing by 3, circle groups ofin the model above. Houghton Mifflin Harcourt Publishing CompanyC How many circles did you draw?Therefore, 15 3 .Reflect1.Make a Conjecture Using the Explore Activity above, make a conjectureabout how to model a fraction division problem.Getting Ready 4.284C

Using Models to Divide MixedFractionsYou can use a model to show division with mixed fractions the same way youmodeled division with whole numbers.EXAMPLECOMMONCORE6.NS.1Model the division expression and find the quotient.1 233 3STEP 1Model the dividend. To model 3 13 draw four rectangles of equal size.Then shade 3 13 of the rectangles.STEP 2Circle groups of 23, which is groups of two 13 -pieces.12345There are 5 groups of 23. So, 3 13 23 5.PracticeModel each fraction division expression, then find the quotient.4 1 2.6 68 1 3.2 284DUnit 2 Houghton Mifflin Harcourt Publishing Company2 2 1. 24 4

LESSON4.2 Dividing Fractions?COMMONCORE6.NS.1Interpret and computequotients of fractions, ,e.g., by using visual fractionmodels .ESSENTIAL QUESTIONHow do you divide fractions?COMMONCOREEXPLORE ACTIVITY 16.NS.1Modeling Fraction DivisionIn some division problems, you may know a number of groups and need tofind how many or how much are in each group. In other division problems,you may know how many there are in each group and need to find thenumber of groups.A You have 34 cup of salsa for making burritos. Eachburrito requires 18 cup of salsa. How many burritoscan you make?To find the number of burritos that canbe made, you need to determine howmany 18 -cup servings are in 34 cup.Use the diagram. How many eighths34are there in 34 ? Houghton Mifflin Harcourt Publishing CompanyYou have enough salsa to makeburritos.18B Five people share 12 pound of cheeseequally. How much cheese does eachperson receive?To find how much cheese each personreceives, you need to determine howmuch is in each ofparts.How much is in each part?Each person will receivepound.Reflect1. Write the division shown by each model.Lesson 4.285

ReciprocalsAnother way to divide fractions is to use reciprocals. Two numbers whoseproduct is 1 are reciprocals.Math On the Spotmy.hrw.com312 43 14123 and 4 are reciprocals.43To find the reciprocal of a fraction, switch the numerator and denominator.numerator ·denominator 1denominatornumeratorEXAMPLE 1COMMONCOREPrep for 6.NS.1Find the reciprocal of each number.A 2992Switch the numerator and denominator.The reciprocal of 29 is 92.Math TalkMathematical PracticesHow can you checkthat the reciprocal in A iscorrect?B 1881Switch the numerator and denominator.The reciprocal of 18 is 81, or 8.C 55 5151Rewrite as a fraction.15Switch the numerator and the denominator.The reciprocal of 5 is 15.Reflect3. Communicate Mathematical Ideas Does every number have areciprocal? Explain.4. The reciprocal of a whole number is a fraction withnumerator.in theYOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com86Unit 2Find the reciprocal of each number.75. 86. 917. 11 Houghton Mifflin Harcourt Publishing Company2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.

EXPLORE ACTIVITY 2COMMONCORE6.NS.1Using Reciprocals to FindEquivalent ValuesA Complete the table below.DivisionMultiplication6 2 37 76 7 7 25 3 58 8 35 8 8 31 5 16 6 561 6 51 1 34 3 41 3 4 1B How does each multiplication problem compare to its correspondingdivision problem?C How does the answer to each multiplication problem compare to theanswer to its corresponding division problem? Houghton Mifflin Harcourt Publishing CompanyReflect8. Make a Conjecture Use the pattern in the table to make a conjectureabout how you can use multiplication to divide one fraction by another.9. Write a division problem and a corresponding multiplication problem likethose in the table. Assuming your conjecture in 8 is correct, what is theanswer to your division problem?Lesson 4.287

Using Reciprocals to Divide FractionsDividing by a fraction is equivalent tomultiplying by its reciprocal.Math On the Spotmy.hrw.com1 14 455EXAMPLE 241 4515COMMONCORE6.NS.1Divide 59 23 . Write the quotient in simplest form.STEP 1Rewrite as multiplication, using the reciprocal of the divisor.5 2 59 329 3AnimatedMathSTEP 2my.hrw.com2The reciprocal ofis 32 .3Multiply and simplify.515 32 918 565 23 569Multiply the numerators. Multiply the denominatorsWrite the answer in simplest form.15 3 518 3 6YOUR TURNPersonalMath TrainerOnline Assessmentand InterventionDivide.911. 35 10910. 25 10my.hrw.comFind the reciprocal of each fraction. (Example 1)1. 252. 19103.3Divide. (Explore 1, Explore 2, and Example 2)4. 43 53 ?ESSENTIAL QUESTION CHECK-IN7. How do you divide fractions?8835. 45 10Unit 26. 12 25 Houghton Mifflin Harcourt Publishing CompanyGuided Practice

NameClassDate4.2 Independent PracticeCOMMONCORE6.NS.18. Alison has 12 cup of yogurt for making fruitparfaits. Each parfait requires 18 cup ofyogurt. How many parfaits can she make? Houghton Mifflin Harcourt Publishing Company Image Credits: Irochka/Fotolia9. A team of runners is needed to run a 14 -mile1relay race. If each runner must runmile,16how many runners will be needed?PersonalMath Trainermy.hrw.comOnlineAssessment andIntervention13. Jackson wants to divide a 34 -pound box oftrail mix into small bags. Each of the bags1will holdpound of trail mix. How many12bags of trail mix can Jackson fill?14. A pitcher contains 23 quart of lemonade.If an equal amount of lemonade ispoured into each of 6 glasses, how muchlemonade will each glass contain?10. Trevor paints 16 of the fence surrounding hisfarm each day. How many days will it takehim to paint 34 of the fence?15. How many tenths are there in 45 ?11. Six people share 35 pound of peanutsequally. What fraction of a pound ofpeanuts does each person receive?16. You make a large bowl of salad to sharewith your friends. Your brother eats 13 of itbefore they come over.112. Biology If one honeybee makes12teaspoon of honey during its lifetime, howmany honeybees are needed to make 12teaspoon of honey?a. You want to divide the leftover saladevenly among six friends. Whatexpression describes the situation?Explain.b. What fractional portion of the originalbowl of salad does each friend receive?Lesson 4.289

FOCUS ON HIGHER ORDER THINKINGWork Area17. Interpret the Answer The length of a ribbon is 34 meter. Sun Yi needspieces measuring 13 meter for an art project. What is the greatest numberof pieces measuring 13 meter that can be cut from the ribbon? How muchribbon will be left after Sun Yi cuts the ribbon? Explain your reasoning.918. Represent Real-World Problems Liam hasgallon of paint for painting101the birdhouses he sells at the craft fair. Each birdhouse requiresgallon20of paint. How many birdhouses can Liam paint? Show your work.19. Justify Reasoning When Kaitlin divided a fraction by 12, the result wasa mixed number. Was the original fraction less than or greater than 12 ?Explain your reasoning.121. Make a Prediction Susan divides the fraction 58 by. Her friend Robyn1651divides 8 by 32 . Predict which person will get the greater quotient. Explainand check your prediction.90Unit 2 Houghton Mifflin Harcourt Publishing Company20. Communicate Mathematical Ideas The reciprocal of a fraction less than1 is always a fraction greater than 1. Why is this?

Going Further4.2COMMONCOREInterpret and computequotients of fractions, andsolve word problems involvingdivision of fractions byfractions, e.g., by using visualfraction models and equationsto represent the problem.Real-World Division?6.NS.1ESSENTIAL QUESTIONHow do fraction division equations relate to the real world?EXPLORE ACTIVITYCOMMONCORE6.NS.1Using Division in a Real-World SituationWhen you are given a division problem, you can create a real-world situation tohelp understand what is going on in the problem. Any fractional amounts inthe problem must make sense in the real-world situation.Write a real-world situation for each division equation.A 23 3 29The equation represents the number of 3s you can divide 23 into, which is 29.A can of paint that is full is shared bypeople. Each persongets of the can of paint.B 6 13 23 9 12 Houghton Mifflin Harcourt Publishing CompanyThe equation represents the number of 23 s you can divide 6 31 into,which is.Marie is filling flower vases with water. She has a pitcher that holdscups of water, and puts cup in each vase. Marie can fillvases.Going Further 4.290A

PracticeComplete the real-world situation for each division equation.1. 34 12 1 12It takes hour to build a birdhouse. If John works for hour today,he can buildbirdhouses.3 102. 1 78 16Tom has a bottle of juice that contains-quart servings. He can pourquarts and is pouringservings.Write a real-world situation for each division equation.4. 54 12 2 1290BUnit 2 Houghton Mifflin Harcourt Publishing Company3. 12 26 13 37

Game4.2INSTRUCTIONSPlaying the GameSTEP 112One student or the teacher is the caller, andgets the caller cards. Each caller card contains afraction problem and its answer.3415471799292318891357371627494 – 225672979354529567814Each player gets a Fracto! board and 20–25 counters. The center is a freespace. Players should cover the center square before play begins.STEP 3On each turn, the caller reads aloud a problem to the players. The callerdoes not read the answer aloud.STEP 4Solve the problem read by the caller. Then search for the answer on yourFracto! board. If the answer is on your board, cover it with a counter. Houghton Mifflin Harcourt Publishing CompanySTEP 222 4 -999STEP 557172749898796729354578The caller places the card in a discard pile. Play continues until thereis a winner.Game 4.290C

Winning the GameA player who covers five squares in a row horizontally, vertically, ordiagonally says, “Fracto!” The caller uses the cards in the discard pile tocheck that this player has calculated correctly. If so, that player is thewinner. If not, play continues until someone else says, “Fracto!”2453715471816671253 Houghton Mifflin Harcourt Publishing Company!90DUnit 2

Dividing MixedNumbersLESSON4.3?COMMONCORE6.NS.1Interpret and computequotients of fractions,and solve word problemsinvolving division of fractionsby fractions .ESSENTIAL QUESTIONHow do you divide mixed numbers?COMMONCOREEXPLORE ACTIVITY6.NS.1Modeling Mixed Number DivisionAntoine is making sushi rolls. He has 2 12 cups of rice and will use 14 cupof rice for each sushi roll. How many sushi rolls can he make?A To find the number of sushi rolls that can be made, youneed to determine how many fourths are in 2 12. Use fractionpieces to represent 2 12 on the model below.114141211414141414141414B How many fourths are in 2 12? Houghton Mifflin Harcourt Publishing Company Image Credits: Stockbyte /GettyImagesAntoine has enough rice to makesushi rolls.Reflect1.Communicate Mathematical Ideas Which mathematical operationcould you use to find the number of sushi rolls that Antoine can make?Explain.2.Multiple Representations Write the division shown by the model.3.What If? Suppose Antoine instead uses 18 cup of rice for each sushi roll.How would his model change? How many rolls can he make? Explain.Lesson 4.391

Using Reciprocals to DivideMixed NumbersMath On the Spotmy.hrw.comMy NotesDividing by a fraction is equivalent to multiplying by its reciprocal. You canuse this fact to divide mixed numbers. First rewrite the mixed numbers asfractions greater than 1. Then multiply the dividend by the reciprocal of thedivisor.EXAMPLE 1COMMONCORE6.NS.1One serving of Harold’s favorite cereal contains 1 25 ounces. How manyservings are in a 17 12 -ounce box?STEP 1Write a division statement to represent the situation.17 12 1 25STEP 2You need to find how many2117groups of 12.5 are inRewrite the mixed numbers as fractions greater than 1.35 7517 12 1 25 2Rewrite the problem as multiplication using the reciprocal ofthe divisor.3535 75 5722STEP 475The reciprocal of5 is 7 .Multiply.355 5 35 5772215 5 2 125, or 12 12 2Simplify first using the GCF.Multiply numerators. Multiply denominators.Write the result as a mixed number.There are 12 12 servings of cereal in the box.Reflect92Unit 24.Analyze Relationships Explain how can you check the answer.5.What If? Harold serves himself 1 12 -ounces servings of cereal eachmorning. How many servings does he get from a box of his favoritecereal? Show your work. Houghton Mifflin Harcourt Publishing CompanySTEP 3

YOUR TURN6.Sheila has 10 12 pounds of potato salad. She wants to divide the potatosalad into containers, each of which holds 1 14 pounds. How many containersdoes she need? Explain.PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comSolving Problems Involving AreaRecall that to find the area of a rectangle, you multiply length width. If youknow the area and only one dimension, you can divide the area by the knowndimension to find the other dimension.Math On the Spotmy.hrw.comEXAMPL 2EXAMPLECOMMONCORE6.NS.1The area of a rectangular sandbox is 56 23 square feet. The length of thesandbox is 8 12 feet. What is the width?STEP 1Write the situation as a division problem.56 23 8 12STEP 2Math TalkRewrite the mixed numbers as fractions greater than 1.170 1756 23 8 12 32 Houghton Mifflin Harcourt Publishing CompanySTEP 3Rewrite the problem as multiplication using the reciprocal ofthe divisor.1701702 17 17 32310 2 1703 1720, or 6 23 3Mathematical PracticesExplain how to findthe length of a rectanglewhen you know the areaand the width.Multiply numerators. Multiply denominators.1Simplify and write as a mixed number.The width of the sandbox is 6 23 feet.Reflect7.Check for Reasonableness How can you determine if your answeris reasonable?Lesson 4.393

YOUR TURNPersonalMath TrainerOnline Assessmentand Intervention8.The area of a rectangular patio is 12 38 square meters.The width of the patio is 2 34 meters. What is the length?9.1The area of a rectangular rug is 1412 square yards.The length of the rug is 4 13 yards. What is the width?my.hrw.comGuided PracticeDivide. Write each answer in simplest form. (Explore Activity and Example 1)1. 4 14 342. 1 12 2 143 4442423. 4 1 18 4. 3 15 1 17 5. 8 13 2 12 6. 15 13 3 56 7. A sandbox has an area of 26 square feet, andthe length is 5 21 feet. What is the width of thesandbox?8. Mr. Webster is buying carpet for an exerciseroom in his basement. The room will have anarea of 230 square feet. The width of the roomis 12 12 feet. What is the length?ESSENTIAL QUESTION CHECK-IN9. How does dividing mixed numbers compare with dividing fractions?94Unit 2 Houghton Mifflin Harcourt Publishing CompanyWrite each situation as a division problem. Then solve. (Example 2)

NameClassDate4.3 Independent PracticeCOMMONCOREPersonalMath Trainer6.NS.1OnlineAssessment andInterventionmy.hrw.com10. Jeremy

Math Trainer Online Assessment and Intervention Personal my.hrw.com Math On the Spot my.hrw.com Multiplying Fractions and Whole Numbers To multiply a fraction by a whole number, you rewrite the whole number as a fraction and multiply the two fractions. Remember to use the GCF to wri