Problem Solving Use The Distributive Property
LESSON5.3Problem Solving Usethe Distributive PropertyFOCUSCOHERENCERIGORLESSON AT A GLANCEF C R Focus:Common Core State StandardsLearning Objective3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 in therange 10–90 (e.g., 9 80, 5 60) using strategies based on place valueand properties of operations.Solve multiplication problems by using thestrategy draw a diagram.Also 3.OA.A.3, 3.OA.B.5, 3.OA.C.7Language ObjectiveMATHEMATICAL PRACTICESMP1 Make sense of problems and persevere in solving them. MP3 Construct viable argumentsand critique the reasoning of others. MP4 Model with mathematics. MP7 Look for and makeuse of structure.Students use the strategy draw a diagram toexplain how to multiply with multiples of 10.F C R Coherence:MaterialsMathBoardStandards Across the GradesBeforeGrade 3 After2.OA.C.3 3.NBT.A.3 4.NBT.B.5F C R Rigor:Level 1: Understand Concepts.Share and Show (Checked Items)Level 2: Procedural Skills and Fluency.On Your OwnLevel 3: Applications.Think Smarter and Go DeeperFC RFor more about how GO Math! fosters Coherencewithin the Content Standards and Mathematical Progressionsfor this chapter, see page 259H.About the MathProfessional DevelopmentUsing an Area Model to MultiplyAn area model helps students visualize multiplication andhow the Distributive Property can be used to multiply withgreater numbers.In this lesson, students learn that they can use theDistributive Property and draw a corresponding areamodel as strategies to multiply with greater numbers(multiples of 10). For example, to find 3 20, theDistributive Property can be used to break apart the factor20 into 10 10. So, 3 20 3 (10 10). Students thenshade 3 rows of 10 squares adjacent to another group of3 rows of 10 squares.Students connect the model to the Distributive Property asthey find the product of each smaller rectangle, and thenadd the partial products to find the total product.Professional Development Videos275AChapter 53 20 3 (10 10)1103103003003 10 303 10 3030 30 603 20 60
1 ENGAGEDaily RoutinesCommon CoreProblem of the Day 5.3Use the factors 3 and 6 to write an equationthat shows the Commutative Property ofMultiplication. 6 3 3 6Vocabularywith the Interactive Student EditionEssential QuestionHow can you use the strategy draw a diagram to multiplywith multiples of 10?Making ConnectionsInvite students to tell you what they know about multiples of 10.What is a multiple of 10? a number that has 10 as a factor Whatpattern do you see in multiples of 10? Possible answer: They endin 0. What digit does 30 end in? 0 Interactive Student Edition Multimedia eGlossaryLearning ActivityFluency BuilderCommon Core FluencyStandard 3.NBT.A.2Add 3-Digit Numbers Write the followingproblems on the board. Have studentspractice adding 3-digit numbers by solvingthe problems and checking their answerswith a partner.196 235 431308 692 1,000217 384 601639 257 896435 321 756972 26 998What is the problem the students are trying to solve? Connect thestory to the problem. How many boxes are there? 3 boxes How many golf balls are in each box? 30 golf balls What operations could you use to find the total number of golfballs? multiplication, addition What equations could you use to find the total number of golfballs? 3 30, 30 30 30Literacy and MathematicsChoose one or more of the following activities. Ask students to clarify the story and the problem by asking themto pick out the important information from the lesson opener. Have students think of reasons a golfer might hit so many golfballs. Why did he want to practice?How can you use thestrategy draw a diagramto multiply with multiplesof 10?Lesson 5.3275B
LESSON5.32 EXPLORE3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 80, 5 60) using strategiesbased on place value and properties of operations.PROBLEM SOLVINGLesson 5.3NameUnlock the ProblemProblem Solving Use the Distributive PropertyMATHEMATICAL PRACTICESEssential Question How can you use the strategy draw a diagram to multiplywith multiples of 10?MP4 Model with mathematics. Make surestudents understand that they need to findthe number of third graders at the assembly.Discuss with students how the graphicorganizer helps to organize their problemsolving. Why are you using the DistributiveProperty? Possible answer: if I break apart the factorUnlockUnlock thethe ProblemProblemThe school assembly room has 5 rows of chairs with 20 chairsin each row. If the third-grade classes fill 3 rows of chairs, howmany third graders are at the assembly?What do I need to find?third gradersI need to find how manyare at the assembly.Discuss with students that drawing a modelon grid paper is like making an array. Itshows rows and columns but there is no spacebetween the rows and columns. This model iscalled an area model. How does the diagram show theDistributive Property? Possible answer: it showsElicit Prior Knowledge101103030What information do I needto use?20 chairs in each row.There are3 rows of chairs.The third graders fillI can use the sum of the products of thesmaller rectangles to find how manythird graders are at the assembly.303 10 How will I use the information?The Distributive Property tells me I can Houghton Mifflin Harcourt Publishing CompanyELLStrategy:Draw a diagram. Finish the shading toshow 3 rows of 20 chairs.33 rows of 10 added to another 3 rows of 10.I could break apart the 3 into a 1 and a 2. Multiply2 20 40 and 1 20 20. Add 40 20 60.Solve the ProblemRead the Probleminto smaller numbers, I can multiply by smaller numbers.MP4 Model with mathematics. If the third graders fill 3 rows, how manyseats are left over? 40 seats Could you break apart the factors in adifferent way? Explain. Yes; possible explanation:Number and Operations in BaseTen—3.NBT.A.3 Also 3.OA.A.3,3.OA.B.5, 3.OA.C.7MATHEMATICAL PRACTICESMP1, MP3, MP4, MP7break apart the factor 20 to multiply.10 )3 20 3 (10 303 10 30 6030 603 20 60 third graders are atSo,the assembly.1. Explain how breaking apart the factor 20 makes finding thePossible explanation: I can use facts I know to find the product.product easier.Chapter 5 275Students can learn to multiply by 10s using thestrategies they have used for other numbers. Select a problem from the lesson. Havestudents draw an array or rectangle torepresent the problem. Show students how to solve theproblem by breaking apart the largerrectangle into smaller ones. For example,4 30 4(10 10 10). Have students use their models to explain toa partner how they arrived at their answers. Remind them that the breaking apartstrategy is also called the DistributiveProperty. Encourage them to use that termin their explanation.DifferentiatedInstruction3Reteach 5.3Enrich 5.321Lesson 5.3ReteachNameLesson 5.3EnrichNameApply the Distributive PropertyProblem Solving Use theDistributive PropertyUse the Distributive Property to help solve each problem.There are 6 rows of singers in a performance. Thereare 20 singers in each row. How many singers are inthe performance?Read the ProblemSolve the ProblemWhat do I need to find?10in the performanceEach row has 606What information do I need to use?6101.How much money does the artistmake on these sales in all?2.The artist sells 2 more paintingsand 4 more sculptures at the sameprices. What is the total amount ofmoney the artist has made so far?3.How many more paintings,sculptures, and photographs wouldthe artist need to sell to makeanother 500?Next, I break apart 20 into 10 10and find the products of the two smallerrectangles.I can draw a diagram and use theDistributive Property to break apart thefactor 20 into 10 1 10 to use facts Iknow.6 3 10 5606 3 10 56060 5 120120So, there are 120 singers.16 3 20 5Eight teams play in a Little Leagueseries. Each team has 20 players.How many players are in theseries?2.How many stickers does Lee havein all?5.Lee gives 4 sheets with 20 stickersand 3 sheets with 10 stickers to hersister. How many stickers does Leehave left?6.Now Lee gives some stickers toher friend Myla. What sheets doesLee give to Myla if she has 200stickers left?430 stickers 640Then, I find the sum of the twoproducts.604. 360First, I draw and label a diagram toshow 6 rows of 20 singers.singers.How will I use the information?1.Lee has 6 sheets of stickers with30 stickers on each sheet. She has8 sheets with 20 stickers each and9 sheets with 10 stickers each.60rows of singers.20Use this problem for 4–6.An artist sells 4 paintings for 20 each, 4 sculptures for 60 each, and 4 photographs for 10 each at her art show.Record the steps you used to solvethe problem.I need to find how many singers areThere areUse this problem for 1–3.320 stickersPossible answer: 5 paintings,Possible answer: 2 sheets with5 sculptures, 10 photographs30 stickers and 3 sheets with20 stickersThe assembly room has 6 rowswith 30 chairs in each row. If thirdgraders fill 3 rows, how many thirdgraders are in the room?7.How did the Distributive Property helpyou solve the problems?Possible answer: I used the Distributive Property to break apartgreater numbers into lesser numbers that are easier to work with.275 Chapter 5160 playersChapter Resources Houghton Mifflin Harcourt Publishing Company90 third graders5-9ReteachChapter Resources Houghton Mifflin Harcourt Publishing Company5-10Enrich
Try Another ProblemFor this problem, make sure studentsunderstand that they need to find the numberof people marching in the band. Have studentsanswer the questions in the graphic organizerand solve the problem. Invite students to sharetheir diagrams by sketching them on the board.Ask them to communicate the steps they used. What strategy and steps did you use tosolve the problem? Possible answer: I broke apartTry Another ProblemMegan is watching a marching band practice.The band marches by with 4 rows of peopleplaying instruments. She counts 30 peoplein each row. How many people march inthe band?Read the Problemthe factor 30 into 10 10 10. I shaded 4 rows of10. I used the diagram to find the products of thesmaller rectangles. Then I added the products.Solve the ProblemWhat do I need to find?Record the steps you used to solvethe problem.I need to find how manypeople march in the band.10411014040 Why do you add the products of thesmaller rectangles to find the totalproduct? Possible answer: each smaller product is1040part of the total product. I add the parts together tofind the whole product.What information do Ineed to use? How could you break apart the factor 30into three addends? Possible answer: I couldThe band has 4 rows of people.There are 30 people in each row.How will I use theinformation?I will draw a diagram and breakapart the factor 30 into 10 10 10 to use facts I know.So, 120 people march in the band.2. How can you check to see if your answer is reasonable?Possible answer: there are 4 rows with 30 people in each row; 30 30 30 30 120.So, my answer of 120 people is reasonable.3. Explain how you can use the Distributive Property to help youfind a product.Possible explanation: I can break apart the greater factor to use facts Iknow. Then I can add the products to find the total product.276break apart 30 into 10 10 10. Houghton Mifflin Harcourt Publishing Company Image Credits: Visions of America, LLC/Alamy ImagesFirst, I shade 4 rows of 30, or 10 10 10.Next, I find the products of the three smallerrectangles.4 10 40 4 10 40 4 10 40Then, I find the sum of the three products.40 40 40 1204 30 120 What is another way you can break apartthe factor 30? Possible answer: 10 20MP2 Reason quantitatively andabstractly. How can knowing how to use theDistributive Property help you multiplywhen you are making a purchase?You may suggest that students placecompleted Try Another Problemgraphic organizers in their portfolios.COMMON ERRORSError Students may use addition instead ofVisual / KinestheticIndividual / PartnersAdvanced LearnersMaterials Base-Ten Grid Paper (see eTeacher Resources) Have students find all the possible ways to break apart afactor into multiples of 10 for 4 60 . Students shouldbreak apart 60 into addends that are multiples of 10. Letstudents know that they may use any number of addends. Have students show each example on grid paper with anequation that shows the Distributive Property.410 10 10 10 10 104040404040404 x 60 4 x (10 10 10 10 10 10)430120 301204 x 60 4 x (30 30) Have studentswrite anexplanation forhow they foundall the ways tobreak apart thefactor.multiplication when using an area model tosolve a problem.Example To solve Try Another Problemstudents write:4 10 14 4 10 144 10 1414 14 14 424 30 42Springboard to Learning Remind studentsthat an area model shows equal groups. Helpstudents recognize that these groups areshown by rows, which each have the samenumber of squares. Point out that the numberof rows and the number of squares in eachrow represent factors.Lesson 5.3276
DO NOT EDIT--Changes must be made through “File info”CorrectionKey AName3 EXPLAINShareShare andand ShowShowUnlock the ProblemMATH Circle the numbers you will use. Use the Distributive PropertyBOARD1. People filled all the seats in the front section ofShare and Showand break apart a greater factorto use facts you know.the theater. The front section has 6 rows with40 seats in each row. How many people are inthe front section of the theater?MATHBOARDThe first problem connects to the learningmodel. Have students use the MathBoard toexplain their thinking. Suggest that studentsbegin solving the problem by first writing theproblem they need to solve (6 40).Exercise 2 requires students to reinterpretExercise 1 given different information.MP3 Construct viable arguments andcritique the reasoning of others. Havestudents compare their models in Exercise 1and discuss different ways to break apart thefactors.Use the checked exercises for Quick Check.Students should show their answers for theQuick Check on the MathBoard. Draw a diagram to help yousolve the problem.First, draw and label a diagram to break apartthe problem into easier parts to solve.10160610160101601060Next, find the products of the smaller rectangles.6 6010 6 6010 10 6 606010 6 Then, find the sum of the products.24060 60 60 60 240So, there arepeople in the front section of the theater.IfThen3322112. What if seats are added to the front section of the theaterRt Ia student misses the checkedexercisesDifferentiate Instruction with Reteach 5.3 Personal Math Trainer 3.NBT.A.3 RtI Tier 1 Activity (online)so that there are 6 rows with 50 seats in each row? Howmany seats are in the front section? Houghton Mifflin Harcourt Publishing CompanyQuick Check300 seatsOnOn YourYour OwnOwn3.SMARTERTova sewed 60 pieces of blue ribbon together to makea costume. Each piece of ribbon was 2 meters long. She also sewed40 pieces of red ribbon together that were each 3 meters long. DidTova use more blue ribbon or red ribbon? Explain.Chapter 5 Lesson 3On Your OwnIf students complete the checked exercisescorrectly, they may continue with the On YourOwn section.SMARTERMath on the SpotVideo TutorUse this video to help students model andsolve this type of Think Smarter problem.Math on the Spot videos are in the InteractiveStudent Edition and at www.thinkcentral.com.277 Chapter 5She used the same amount. 60 2 120; 40 3 1203 MNLESE342118 C05L03.indd 27727726/02/14 5:47 PM
MATHEMATICAL PRACTICES4.ANALYZE t -00, '03 4536 563& t 13& *4*0/MATHEMATICAL3 Verify the Reasoning of OthersPRACTICECarina draws this diagram to show that 8 30 210.Explain her error.Possible explanation:1011104 ELABORATE10MATHEMATICAL PRACTICES8MP3 Construct viable arguments andcritique the reasoning of others. InExercise 4, students need to analyze thediagram to find the error.Carina shaded 7 rows of 30 and added the products 7 10 70,7 10 70, and 7 10 70; 70 70 70 210. She shouldDEEPERhave shaded 8 rows of 30 and added the products 8 10 80,Exercise 6 is a multistep problem thatinvolves addition and subtraction. Suggeststudents write a number sentence torepresent the problem.8 10 80, and 8 10 80; 80 80 80 240. So, 8 30 240.5.Math Tamika wants to display 10 trophies on aWRITEtable in a rectangular array. How many different ways canTamika arrange the trophies? Explain your answer.SMARTER4 ways; possible explanation: she can make 10 rows of 1,Students will need to identify applications ofthe Distributive Property. Each of the correctanswers shows different representations ofequations that use the Distributive Property.Students should be able to recognize theproperty’s application in different formats.1 row of 10, 2 rows of 5, or 5 rows of 2.6.DEEPERThe drama club has 350 tickets to sell. They sell124 tickets on Monday and 98 tickets on Tuesday. Howmany tickets does the drama club have left to sell?128 ticketsSMARTERSelect the equations that show theDistributive Property. Mark all that apply. Houghton Mifflin Harcourt Publishing Company7.A 3 20 (3 10) (3 10)B (7 3) 8 7 (3 8)C (5 10) (5 10) 5 20D (9 2) (9 4) 9 65 EVALUATE FormativeAssessmentEssential QuestionUsing the Language ObjectiveReflect Have students draw and explain toanswer the Essential Question.How can you use the strategy draw adiagram to multiply with multiples of 10?278DIFFERENTIATED INSTRUCTIONDINDEPENDENT ACTIVITIESPossible answer: I can draw and shade a rectangle ongrid paper to show the problem. Then I can break aparta factor to make smaller rectangles for facts I know.Math JournalDifferentiated Centers KitActivitiesMultiplicationDashLiteratureParty Plans bythe Numbers!GamesMultiplicationBingoWRITEMathWrite a description of how a diagram canhelp you solve 2 40.(BNFTStudents completeorange ActivityCard 17 by usingmodels to applymultiplication factsthrough 12 by 12.Students readthe book and usemultiplication factsand strategies toplan a party.Studentspracticemultiplicationfacts through 10.Lesson 5.3278
Practice and HomeworkLesson 5.3NameProblem Solving Use theDistributive PropertyPractice and HomeworkCOMMON CORE STANDARD—3.NBT.A.3Use place value understanding and properties ofoperations to perform multi-digit arithmetic.Read each problem and solve.1. Each time a student turns in a perfect spelling test,Use the Practice and Homework pages toprovide students with more practice of theconcepts and skills presented in this lesson.Students master their understanding as theycomplete practice items and then challengetheir critical thinking skills with ProblemSolving. Use the Write Math section todetermine student’s understanding of contentfor this lesson. Encourage students to use theirMath Journals to record their answers.Ms. Ricks puts an achievement square on the bulletinboard. There are 6 rows of squares on the bulletin board.Each row has 30 squares. How many perfect spellingtests have been turned in?Think: 6 30 6 (10 10 10) 60 60 60 180180 spelling tests2. Norma practices violin for 50 minutes every day.How many minutes does Norma practice violin in7 days?350 minutes3. A kitchen designer is creating a new backsplash forthe wall behind a kitchen sink. The backsplash willhave 5 rows of tiles. Each row will have 20 tiles.How many tiles are needed for the entire backsplash?100 tiles Houghton Mifflin Harcourt Publishing Company4. A bowling alley keeps shoes in rows of cubbyholes.There are 9 rows of cubbyholes, with 20 cubbyholesin each row. If there is a pair of shoes in every cubbyhole,how many pairs of shoes are there?180 pairs of shoes5.Math Write a description of how a diagramWRITEcan help you solve 2 40.Check students’ work.Chapter 5PROFESSIONALDEVELOPMENTMathematical Practices in Your ClassroomCCSS.Math.Practice.MP1 Make sense of problemsand persevere in solving them.Students can persevere in solving problems involving multiplyingwith multiples of 10 by breaking apart a factor and drawing acorresponding area model. Students make a plan for breakingapart a factor that makes multiplying easier. By drawing an areamodel, students also learn how to visualize situations involvingequal groups.279 Chapter 5279The following activities can help promote students’ use of breakingapart a factor and area models as entry points to solving problemsinvolving equal groups with multiples of 10: Draw an area model showing the number of months in the yearthat have 30 days to find the total number of days. Draw an area model showing how many stamps you have if youbuy 6 sheets of stamps with 20 stamps on each sheet. Write a word problem about a garden with a number of rowswith 40 plants in each row.
Lesson Check (3.NBT.A.3)1. Each snack pack holds 20 crackers.How many crackers in all are there in4 snack packs?2. A machine makes 70 springs eachhour. How many springs will themachine make in 8 hours?Continue concepts and skills practice withLesson Check. Use Spiral Review to engagestudents in previously taught concepts andto promote content retention. Common Corestandards are correlated to each section.560 springs80 crackersSpiral Review (3.OA.A.1, 3.NBT.A.1, 3.MD.B.4)3. Lila read 142 pages on Friday and168 pages on Saturday. Estimate howmany pages Lila read on Friday andSaturday combined.Possible answer: about 300 pages4. Jessica wrote 6 6 6 6 on theboard. What is another way to show6 6 6 6?Possible answers: 4 6 or 6 4Use the line plot for 5–6.number of birds he saw at his birdfeeder. How many more sparrows thanblue jays did he see?4 more sparrows6. How many robins and cardinalscombined did Eliot see? Robin Blue Jay Sparrow CardinalBirds Seen5 robins and cardinals280 Houghton Mifflin Harcourt Publishing Company5. Eliot made a line plot to record theFOR MORE PRACTICEGO TO THEPersonal Math TrainerLesson 5.3280
How did the Distributive Property help you solve the problems? Lesson 5.3 Enrich Apply the Distributive Property Use the Distributive Property to help solve each problem. Possible answer: I used the Distributive Property
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the Distributive Property of Multiplication being used when numbers within the parentheses are being subtracted. Example 5(7 4) 3(x 2) Both of these examples would require the use of the Distributive Property. II. Use the Distributive Property
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