Algebra Relate Multiplication And Division

LESSON6.7Algebra Relate Multiplicationand DivisionFOCUSCOHERENCERIGORLESSON AT A GLANCEF C R Focus:Common Core State StandardsLearning Objective3.OA.B.6 Understand division as an unknown-factor problem.Use bar models and arrays to relatemultiplication and division as inverseoperations.Also 3.OA.A.2, 3.OA.A.3, 3.OA.A.4, 3.OA.C.7MATHEMATICAL PRACTICESMP2 Reason abstractly and quantitatively. MP4 Model with mathematics.MP7 Look for and make use of structure. MP8 Look for and express regularityin repeated reasoning.F C R Coherence:Standards Across the GradesBeforeGrade 3 After2.OA.C.4 3.OA.B.6 4.NBT.B.6F 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 DeeperLanguage ObjectiveStudents reread the lesson and rephrase theexplanation on how to use multiplication todivide.MaterialsMathBoardF C R For more about how GO Math! fosters Coherencewithin the Content Standards and Mathematical Progressionsfor this chapter, see page 299J.About the MathProfessional DevelopmentIf Students AskInteractive Student EditionWhat are inverse operations? An inverse operation isone that is the opposite of or undoes another operation.Addition and subtraction are inverse operations.Multiplication and division are inverse operations.Personal Math TrainerOne way for students to model the inverse relationshipbetween multiplication and division is by using an array.The same array can represent a multiplication fact and itsrelated division fact. The array is used in different ways tofind a product or quotient.For example, in an array with 5 equal rows of 3, youcan write 5 3 15 or 15 3 5, depending onthe situation. Students should begin to see how themultiplication and division equations for the same arrayare related.Students also apply the inverse relationship by usingmultiplication facts they know to solve division facts.Professional Development Videos339AChapter 6Math on the SpotAnimated Math ModelsiTools: CountersiTools: Number Charts

1 ENGAGEDaily RoutinesCommon CoreProblem of the Day 6.7An aquarium has 346 fish. Another 433 fishare added to the aquarium. Estimate thetotal number of fish in the aquarium to thenearest 10.780Vocabulary inverse operations Interactive Student Edition Multimedia eGlossarywith the Interactive Student EditionEssential QuestionHow can you use multiplication to divide?Making ConnectionsInvite students to tell you what they know about multiplication.What is the relationship between multiplication and addition?Multiplication is repeated addition. What is the multiplication problemthat represents 4 4 4? 3 4 What is 3 4? 12Learning ActivityWhat is the problem the students are trying to solve? Connect thestory to the problem. How many letters were delivered? 24 How many mailboxes were they delivered to? 6Vocabulary BuilderMaterials index cards What is the problem asking? how many letters were delivered to eachmailboxLiteracy and MathematicsChoose one or more of the following activities.Inverse Operations Give students indexcards labeled inverse operations, addition,subtraction, multiplication, and division.Explain that inverse means opposite.Ask students what inverse operationsmeans. Remind students that addition,subtraction, multiplication, and division areoperations. Students should see that inverseoperations would be opposite operations. Have students draw 4 or 5 mailboxes with an equal number ofletters inside. Have students indicate the number of mailboxes, thenumber of letters in each box, and the total number of letters. Have students draw an array of 4 or 5 groups of 6 items. Havethem count the number of items in the array. Have students writea scenario to go along with the arrays that they created.Ask students to sort the operation cardsto show inverse operations. Studentsshould pair addition with subtraction andmultiplication with division.Literature ConnectionFrom the Grab-and-Go Differentiated Centers KitStudents read abouthow multiplication anddivision are related on amultiplication table.How can you usemultiplication todivide?The Homework TableLesson 6.7339B

LESSON6.72 EXPLORE3.OA.B.6 Understand division as an unknown-factor problem.DO NOT EDIT--Changes must be made through "File info"CorrectionKey BUnlock the ProblemUnlockUnlock thethe ProblemProblemPam went to the fair. She went on thesame ride 6 times and used the samenumber of tickets each time. She used18 tickets. How many tickets did she useeach time she went on the ride?One WayThis method uses bar models to show howmultiplication and division are related. What number goes in each bar? 3 How do you know? Possible answer: I thoughtOne Way“What number times 6 equals 18?” So, I wrote 3 ineach bar. How can you check your answer? PossibleMP6 Attend to precision. It is important forstudents to notice the relationships betweenfactors, products, divisors, and quotients. How is the quotient of the divisionequation related to the unknown factor inthe multiplication equation? They are the same.ELL Strategy:Identify RelationshipsTell students that multiplication and divisionare inverse operations, or opposites, just likeaddition and subtraction are opposites. Inverse means opposite. What other thingsare opposite or inverse? Possible answers: tall/short, big/small Have students describe inverse relationshipswith words and gestures: An elephant isbig. A mouse is small. Big and small areopposites.Use bar models.Complete the bar model to show6 groups of 3 tickets.3333333318 tickets31833tickets18Write: 6 3 3Write: 18 6 3 tickets eachSo, Pam usedtime she went on the ride.Multiplication and division are oppositeoperations, or inverse operations.You can think about multiplication tosolve a division problem.MathTalkTo solve 18 6 , think 6 18.MATHEMATICAL PRACTICES 5Use a Concrete ModelWhat information is givenin each of the bar models?Since 6 3 18, then 18 6 3.Possible answer: the division model shows 18 tickets divided into 6 equal groups of 3 tickets each.Chapter 6 339The multiplication model shows 3 tickets in 6 equal groups equals 18 tickets.3 MNLESE342125 C06L07.indd 33910/7/14 7:17 PM3Reteach 6.7Enrich 6.72DO NOT EDIT--Changes must be made through "File info"CorrectionKey ADO NOT EDIT--Changes must be made through "File info"CorrectionKey A1Lesson 6.7ReteachNameDifferentiatedInstructionLesson 6.7EnrichNameAlgebra Relate Multiplicationand DivisionMultiplication and Division MatchYou can use an array to complete 21 4 3 5Solve. Then draw a line to match each multiplicationequation to a related division equation.Use 21 counters.Make 3 equal rows.There are 7 counters in each row.1.238516 A12 4 2 5 62.538540 B42 4 7 5 63 rows of 7 5 213.339527 C18 4 3 5 6The 21 tells the total number of counters in the array.The 3 stands for the number of equal rows.The 7 stands for the number of counters in each row.4.637542 D40 4 8 5 5You can use a related multiplication fact to check your answer.5.236512 E24 4 6 5 46.537535 F27 4 9 5 37.634524 G24 4 3 5 88.838564 H36 4 9 5 49.336518 I16 4 2 5 810.934536 J18 4 2 5 911.932518 K64 4 8 5 812.833524 L35 4 5 5 7So, 21 4 3 5 721 4 3 5 7 3 3 7 5 21So, 3 rows of 7 represents 21 4 3 5 7 or 3 3 7 5 21.Complete.1.2.6 rows of63424 4 6 545 245 2444.6375 42Chapter Resources Houghton Mifflin Harcourt Publishing Company3 MNLEAN342996 C06R07.indd 173.3 rows of9335 27927 4 3 5Complete the equations.339 Chapter 6each time she went on the ride Circle the numbers you need to use.Complete the bar model to show18 tickets divided into 6 equal groups.3 Houghton Mifflin Harcourt Publishing Company Image Credits: kali9/Getty Imagesrelated.Use Math Talk to focus on howmultiplication and division arethe number of tickets Pam usedWhat if the problem said Pam wenton the ride 6 times and used 3 ticketseach time? How many tickets did Pamuse in all? How do you know when to divide andwhen to multiply to solve a word problem?MathTalk What do you need to find?You can use bar models tounderstand how multiplicationand division are related.answer: I can count by 3s or add 3 until I get 18.and division undo each other. So if you know that4 3 12, you know that 12 divided by 4 is 3.Operations and AlgebraicThinking—3.OA.B.6 Also 3.OA.A.2,3.OA.A.3, 3.OA.A.4, 3.OA.C.7MATHEMATICAL PRACTICESMP3, MP4, MP5, MP6Essential Question How can you use multiplication to divide?MP1 Make sense of problems andpersevere in solving them. Be surestudents know that they are to divide to findthe number of tickets used each time. Why can you use multiplication to solve adivision problem? Possible answer: multiplicationLesson 6.7Relate Multiplication and DivisionMATHEMATICAL PRACTICESPossible answer: if you want to find the total of all theequal groups, multiply. If you want to find the numberin each equal group or how many equal groups, divide.ALGEBRAName42 475 278 rows of839565.6-17864 4 8 59365 5485 645 64854 4659ReteachChapter Resources Houghton Mifflin Harcourt Publishing Company17/02/14 6:50 PM3 MNLEAN342996 C06E07.indd 186-18Enrich17/02/14 6:42 PM

DO NOT EDIT--Changes must be made through "File info"CorrectionKey ADO NOT EDIT--Changes must be made through "File info"HandsCorrectionKey AOnAnother WayThis method uses arrays to show therelationship between multiplication anddivision. How can the array show both a divisionproblem and a multiplication problem?HandsOnAnother Way Use an array.You can use an array to see how multiplication and divisionare related.Show an array with 18 counters in 3 equal rows bycompleting the drawing.Possible answer: arranging 18 counters in 3 rows tofind the number of counters in each row, you divide18 3 6. When you make 3 rows of 6 counterseach and combine them to find how many countersin all, you multiply 3 6 18.The same array can be used tofind the total number if youknow there are 3 rows with 6counters in each row.3 EXPLAIN6 counters in each row.There are6Write: 18 3 ShareShare andand ShowShow18Write: 3 6 MATHBOARDMathTalkN1. Use the array to completeMATHEMATICAL PRACTICES 3Think: There are 3 counters in each row.See below.5 153 rows of62 rows of 1273 rows of 2153 156 122 73 21515 3 612 2 721 3 Complete the equations.8 405. 5 840 5 BOARDMathTalk4.3 186. 6 318 6 Math Talk: Possible description: think of a related multiplication fact:2 6. Since 2 3 6, then 6 2 3.Use Math Talk to focus onstudents’ understanding thatthere are more than one way to solve aproblem. Houghton Mifflin Harcourt Publishing Company3. Houghton Mifflin Harcourt Publishing Company2.MATHThe first problem connects to the learningmodel. Have students use the MathBoardto explain their thinking about howmultiplication can be used to divide.Apply What is anotherway to find thequotient for 6 2?the equation.36 2 Complete.Share and ShowUse the checked exercises for Quick Check.Quick Check321Rt Ia student misses the checkedexercisesIf340Then3 MNLESE342125 C06L07.indd 340AdvancedAdvancedLearnersLearners13/02/14 11:50 AMDifferentiate Instruction with Reteach 6.7 Personal Math Trainer 3.OA.B.6 Rtl Tier 1 Activity (online)Logical / MathematicalIndividual / Partners Have students copy the following equations. Thentell them to fill in each circle with one of thefollowing symbols: , , , or .1. 20 5 5 52. 9 4 40 43. 16 2 10 24. 45 5 9 1 Ask students to write 2 of their own operationexercises like the ones above.COMMON ERRORSError Students may use an array incorrectlyCOMMONERRORSto solvea division problem.Example20 4 4Springboard to LearningHave students count andlabel the number of rowsand columns after theyhave completed the array. 4Encourage them to checkthat they do not confusethe divisor and the quotient.5Lesson 6.7340