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8.2 Associative Property of ? Essential Question How can you use the Associative Property of Multiplication to find products?

Texas Essential Knowledge and Skills and Operations—3.4.F Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding facts; 3.4.K Solve one-step and two-step problems involving multiplication and division within 100 How can you use the Associative Property of Algebraic Reasoning—3.5.B Represent and solve one- Multiplication to find and two-step multiplication and division problems within 100 products? Also 3.4.D, 3.4.G, 3.5.D MATHEMATICAL PROCESSES 3.1.E Create and use representations 3.1.G Display, explain, and justify mathematical ideas and arguments Are You Ready? Access Prior Knowledge Use the Are You Ready? 8.2 in the Assessment Guide to assess students’ Lesson Opener understanding of the prerequisite skills Making Connections for this lesson. Invite students to tell you what they know about different modes of transportation. Explain to students that the Mole Mountain Express described in the problem is like a Vocabulary taxi, shuttle, or bus. Associative Property of Multiplication Have you ever taken a taxi, a shuttle, or a bus? What was it like? Have you ever taken Go to Multimedia eGlossary at bags with you on a trip? How many? thinkcentral.com Using the Digital Lesson You may wish to model the stated problem with students using pictures or manipulatives, such as counters.

Learning Task What is the problem the students are trying to solve? Connect the story to the problem. Resources • How many passengers go each time? (4 passengers each trip) • How many bags does each passenger take? (2 bags per passenger) For the student For the teacher • How could you find the number of bags delivered on one trip? (Multiply the Interactive Digital Management Student Edition Center organizes program number of passengers, 4, by the number of bags per passenger, 2. Or, add the provides students resources by TEKS! number of bags per passenger, 2, 4 times.) with an interactive learning • What expression could you use to find the number of bags delivered on one trip? environment! eTeacher (4 × 2 or 2 + 2 + 2 + 2) Edition Math on the Spot Video Tutor Online Assessment Literacy and System • Have students look for clue words in the problem that indicate that multiplication iTools Virtual might be used to solve it. Manipulatives • Have students write a short story about one of the travelers. Encourage them to Soar to Success Math include details such as what the traveler has in his/her bag and what he/she is doing Online Intervention at Mole Mountain. Lesson 8.2 243A Name Number and Operations—3.4.F, 3.4.K Associative Property Algebraic Reasoning— 8.2 3.5.B Also 3.4.D, 3.5.D of Multiplication MATHEMATICAL PROCESSES ALGEBRA 3.1.E Connect Essential Question Students relate the Associative Property of ? How can you use the Associative Property of to the Associative Property of Multiplication. Multiplication to find products?

Connect You have learned the Associative Unlock the Problem Property of Addition. When the grouping of the Math Idea addends is changed, the sum stays the same. Have students draw a quick picture of the roller Always multiply the inside the parentheses first. coaster train to check their understanding of the (2 + 3) + 4 = 2 + (3 + 4) problem. The Associative Property of Multiplication states The example uses arrays to represent the problem. that when the grouping of the factors is changed, • What does each separate array show? Possible the product is the same. It is also called the Grouping Property of Multiplication. answer: 2 rows of 2 seats each; 2 × 2 Use the Math Idea to remind students to complete UnlockUnlock the Problem the inside the parentheses first. Each car on the roller coaster • Underline what you need to find. After students complete the problem with the new has 2 rows of seats. Each row has grouping, ask the following questions: 2 seats. There are 3 cars in each • Describe the grouping of the seats. • Does it matter which factors you multiply first? train. How many seats are on 3 cars with 2 rows of 2 seats each roller coaster train? No; whichever way I multiply the factors, the product will still be the same. Use an array. • Was one grouping easier than the other for You can use an array You can change the grouping with × × parentheses and the product is the same. finding the final answer? Explain. Possible to show 3 (2 2). × × = ■ explanation: yes; multiplying by 2 is easier for me, 3 × (2 × 2) = ■ (3 2) 2 ( × ) × so 3 2 2 was easier because I multiplied by _6 × 2 = _12 3 × _4 = _12 2 both times. Math Talk So, there are 3 cars with 4 seats in Mathematical Processes each car. Explain why the products Mathematical Processes 3 × (2 × 2) and (3 × 2) × 2 Math Talk There are _12 seats on each roller are the same. coaster train. Use Math Talk to focus on students’ understanding Possible explanation: it does not matter how the factors that changing the grouping of the factors does not are grouped. The products will be the same. change the product. Module 8 243 © Houghton Mifflin Harcourt Publishing Company

Kinesthetic / Verbal English Language Learners ELL Language Support Small Leveled Activities ELPS ELPS 1.A.2, 2.D.2, 2.I.4 Beginning: Activity 8 2.I.4, 3.G.1, 3.H.3 Intermediate: Activity 3 2.D.2, 2.E.3, 3.F.2 Strategy: Creative Grouping Materials: square tiles Advanced: Activity 41 4.F.3, 4.F.8 • Partner advanced English learners with beginning English learners. Advanced High: Activity 43 4.F.8, 4.G.2, 4.G.4 • Have beginning English learners use tiles to show (2 + 3) + 4; then thinkcentral.com Go to for the ELL Activity show 2 + (3 + 4). Guide containing these leveled activities. • Have the advanced English learner help explain each model and how each shows the Associative Property of Addition.

(2 1 3) 1 4 • Beginning English learners should ask for clarification if they do not 243 Module 8 understand a word or phrase. Example Use the Commutative and Associative Properties. You can also change the order of the factors. The product is the same. Example This example shows that you can change the order of × × = ■ × × = ■ (4 3) 2 4 (3 2) the factors and the product will be the same. × × = ■ × × = ■ 4 (3 2) Associative Property 4 (2 3) • How could the Commutative and Associative 4 × _6 = _24 (4 × 2) × 3 = ■ Associative Property Properties be used to make (3 × 8) × 2 easier to solve? Possible answer: change the order and the _8 × 3 = _24 grouping to 8 × (3 × 2) so that you can use a basic fact, 8 × 6, instead of 24 × 2 or 16 × 3. ShareShare and ShowShow • How can multiplication properties help you find products? Possible answer: you can change the grouping 1. Find the product of 5, 2, and 3. Write another way to group the factors. Is the product the same? Why? or the order of the factors to find products of facts you know. (5 × 2) × 3 = 30; Possible answer: I can regroup the factors as 5 × (2 × 3) = 30; the product is the same because the grouping does not change the product. Share and Show Write another way to group the factors. Then find The first problem connects to the learning model. the product. Possible groupings are given for 2–5. Have students use the MathBoard to explain their 2. (2 × 1) × 7 3. 3 × (3 × 4) thinking. 2 × (1 × 7) (3 × 3) × 4 Use the checked exercises for Quick Check. Students

14 36 should show their answers for the Quick Check on the MathBoard. 4. 5 × (2 × 5) 5. 3 × (2 × 6)

(5 × 2) × 5 (3 × 2) × 6 3 2 1 Quick Check 50 36

IF a student misses the checked exercises Algebra Find the unknown factor.

6. 7 × (2 × _4 ) = 56 7. 60 = (2 × _5 ) × 6 8. _8 × (2 × 2) = 32 Differentiate Instruction with THEN RtI Tier 1 Lesson 37 9. 42 = 7 × (2 × _3 ) 10. 8 × (5 × _1 ) = 40 11. 0 = _0 × (25 × 1)

244 © Houghton Mifflin Harcourt Publishing Company Problems Exercises 6–11 require students to extend their thinking to find an unknown factor when 2 factors and a product are given. Remind students to apply the Associative Property to help solve these problems.

Visual Enrich Individual

• Challenge students to fill in the unknown numbers so that the product of the numbers in each row and each column equal 24.

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• Have students create their own grids in which the numbers in each row and each column have the same product.

Go to Go to thinkcentral.com for additional enrichment activities in the Enrich Activity Guide. Lesson 8.2 244 Name Problem Solving Problem SolvingSolving Use the graph for 12–13. Have students read problem 12 and discuss what they Roller Coasters need to find. Students have to read a graph and draw 12. Multi-Step Each car on the Steel 7 a picture to help solve a multiplication problem with Force train has 3 rows with 2 seats 6 in each row. How many seats are 5 three factors. 4 on the train? Draw a quick picture. 3 2 36 seats; Check students’ drawings. Problems Train per Cars 1 0 Problem 13 requires students to interpret a graph to 13. Multi-Step A Kingda Ka train has 4 seats per car, but the last car solve this multistep problem involving , Kingda Ka has only 2 seats. How many seats are Steel Force Mr. Freeze Roller Coaster multiplication, and addition. on one Kingda Ka train? In Problem 14, students use the Commutative and 18 seats Associative Properties and find that changing the order Write Math Show Your Work or grouping of factors does not change the product. 14. Sense or Nonsense? Each week, Kelly works 2 days for 4 hours each day and Go Deeper earns $5 an hour. Len works 5 days for 2 hours each day and earns $4 an hour. Kelly says Have students compare and contrast the they both earn the same amount. Does this Commutative and Associative Properties of statement make sense? Explain.

Multiplication. They should describe how they are Yes; they both earn $40. (2 × 4) × $5 = $40 and the same and how they are different. (5 × 2) × $4 = $40 Students should be able to verbalize the difference: with the Commutative Property, the order of the 15. Representations What number sentence does this array represent? Write another way to factors changes; with the Associative Property, the group the factors. grouping changes while the order of the factors stays the same. They should also recognize that the product stays the same when applying either the Commutative or Associative Property.

4 × (3 × 2) = 24; Possible answer: (4 × 3) × 2 = 24

COMMONC ERRORS ErrorE Students may not change the grouping so they can work with a fact they know. Module 8 • Lesson 2 245

Example 3 × (2 × 6) = 3 × 12 © Houghton Mifflin Harcourt Publishing Company Springboard to Learning Have students look for factors that have a 1-digit product. These combinations will allow students to find the product of another fact.

3 2 RtI Tier 1 Lesson 37 Enrich 37 1

Name Enrich 37 Name LESSON Algebra • Associative Property of 1 37 3.4.F, 3.4.K Multiplication Matching Factors and Products MathM on the Spot OBJECTIVE Use the Associative Property of Multiplication to multiply with three factors. Complete the number sentence in Column A. Then circle You can use the Associative Property of Multiplication the correct product in Column B. In Column C, use 3 VideoV Tutor to multiply with 3 factors. If you change the grouping of factors, factors to write a number sentence for the product in the product remains the same. Column B that is not circled.

Find 4 × (3 × 1). Find (4 × 3) × 1. Possible answers for Column C are given.

Step 1 Start inside the parentheses. Step 1 Start inside the parentheses. Column A Column B Column C Make 3 groups of 1 counter. Make 4 groups of 3 counters. × × Through the Math on the Spot Video Tutor, (3 1) (4 3) 1. (9 3 1) 3 2 5 18 18 14 (2 3 1) 3 7 5 14

Step 2 Multiply by 4, the number Step 2 Multiply by 1, the number 2. 7 3 (2 3 3) 5 42 42 16 3 3 5 students will be guided through an interactive outside the parentheses. Make outside the parentheses. Make (2 4) 2 16 4 groups of the counters in Step 1. 1 group of the counters in Step 1. 3. 8 3 (2 3 2) 5 32 48 32 3 3 5 solving of this type of H.O.T. problem. Use this 4 × (3 × 1) (4 × 3) × 1 8 (2 3) 48 video to also help students solve the H.O.T. 4. (1 3 2) 3 7 5 14 14 36 (2 3 3) 3 6 5 36 problem in the Interactive Student Edition. With 5. 1 3 (3 3 2) 5 6 6 5 (1 3 1) 3 5 5 5 Step 3 Count the total number of Step 3 Count the total number of 6. 3 3 (8 3 1) 5 24 12 24 3 3 5 these videos and the H.O.T. problems, students counters. 12 counters counters. 12 counters (3 2) 2 12 So, 4 × (3 × 1) = 12 and (4 × 3) × 1 = 12. 7. (3 3 2) 3 6 5 36 36 40 (2 3 5) 3 4 5 40 will build skills needed in the TEXAS assessment. Possible groupings are given. Write another way to group the factors. Then find the product. 8. (3 3 3) 3 5 5 45 54 45 (3 3 3) 3 6 5 54 1. (2 × 3) × 2 2. 2 × (4 × 2) 3. 2 × (3 × 1) 9. 3 3 5 27 Math on the Spot videos are in the × × × × × × 9 (3 1) 27 35 (7 3 1) 3 5 5 35 2 (3 2); 12 (2 4) 2; 16 (2 3) 1; 6 10. (7 3 1) 3 4 5 28 60 28 3 3 5 Interactive Student Edition and at Harcourt Publishing Company © Houghton Mifflin 6 (5 2) 60 thinkcentral.com 4. 5 × (7 × 1) 5. 8 × (4 × 1) 6. 2 × (2 × 6) . 11. Explain how you decided which factors to group in Column C. (5 × 7) × 1; 35 (8 × 4) × 1; 32 (2 × 2) × 6; 24 Possible answer: I grouped factors together to use multiplication facts I know. Number and Operations 73 Enrich E37 245 Module 8 © Houghton Mifflin Harcourt Publishing Company Mathematical Processes Model ¥ Reason ¥ Communicate DailyDaily Assessment TaskTask 3 Fill in the bubble for the correct answer choice. You can 2 Daily Assessment Task 1 use an array, area model, or a strip diagram to represent and solve. Can students use the Associative Property of 16. Michael plays a fishing game at the school carnival. He Multiplication to find products? plays 4 times and catches 3 fish each time. For each fish Michael catches, he wins a of 3 toy planes. How many IF THEN toy planes does Michael win? A 18 C 36 NO • Soar to Success Math B 12 D 9 Warm-Up 12.50

17. Emma helps at a farm 2 days each week. She works 4 hours each day and earns $8 for each hour. How YES • Enrich 37 much money does Emma earn each week? • Homework and Practice A $64 C $48 Lesson 8.2 B $40 D $32

18. Multi-Step Mrs. Lee has 25 baskets of books. She wants to put them on the 2 bookcases in her classroom. Each TEXAS Test Prep Coach bookcase has 4 shelves. Each shelf can hold 3 baskets. How many baskets will be left over? Test Prep Coach helps teachers to identify common errors that students can make. A 7 C 10 In the Test Prep exercise, if students selected: B 1 D 5 A They multiplied 3 × 3, then added 2. B They only multiplied 3 × 3. TEXAS Test Prep C They added 2 + 3 + 3. 19. Mark has 2 rows of 3 car models on each of his 3 shelves. How many car models does he have? A 11 Write ? Essential Question Math B 9 How can you use the Associative Property of C 8 Multiplication to find products? Possible answer: D 18 I can change the way the factors are grouped to work with numbers I know, and still get the same answer. 246 © Houghton Mifflin Harcourt Publishing Company

Differentiated Centers Kit

Games

Games Literature Activities Multiplication Here’s What I Do Diamond Derby Bingo Students read about using Students complete Students practice multiplication tables to purple Activity Card multiplication win a computer game 15 by practicing facts through 10. about multiplication. multiplication facts through 10 by 10.

Lesson 8.2 246 5

Number and Operations—3.4.F, 3.4.K Algebraic Reasoning—3.5.B Also 3.4.D, 3.5.D Homework MATHEMATICAL PROCESSES 3.1.E and Practice Lesson CheckCheck TEXAS Test Prep Name 8. Malcolm put stamps on 5 pages 9. Ana works at the library 3 days each of an album. He puts 3 rows of week. She works for 2 hours each day 8.2 2 stamps on each page. How many and earns $7 each hour. How much Associative Property of Multiplication stamps does Malcolm have in his money does Ana earn each week? ALGEBRA album? A $42 Write another way to group the factors. Then find the product. Possible groupings A 10 are given for 1–4. B $21 × × × × 1. (3 2) 5 2. 6 (2 2) B 30 C $12 × × × × 3 (2 5) (6 2) 2 C 20 D $35 30 24 D 25

3. 5 × (2 × 4) 4. (2 × 3) × 6

(5 × 2) × 4 2 × (3 × 6) 10. Zeff, Maggie, and Jani each make 11. There are 6 boxes of paints. Each box 2 kites. Each kite has 4 tails. How has 3 rows of 2 paint cans. Which 40 36 many tails are there on all of the shows how many paint cans in all? kites? Problem SolvingSolving A 6 + (3 × 2) = 12 A 8 × + = Use the pictograph for 5–7. B (6 3) 2 20 Sticker Books B 24 C (6 × 3) × 2 = 18 5. Mario’s sticker books have 4 rows of Name Number of Books C 12 3 stickers in each row. How many D × × = Mario 6 (3 2) 36 stickers are in Mario’s books? D 18 Draw a picture. Tamisha

24 stickers; Pang Check students’ drawings. Key: Each = 1 sticker book 12. Multi-Step Gwen has 30 action 13. Multi-Step Reggie has 65 quilt 6. Pang’s sticker books have 3 rows figures. She puts them in 2 boxes. patches. He makes 2 quilts with of 3 stickers. He uses 5 stickers. Each box holds 3 rows of 4 action 5 rows of 6 patches in each quilt. How How many stickers are still in the sticker books? _____31 stickers _ figures. How many action figures many quilt patches will be left over? will be left over? A 43 7. Tamisha’s sticker books have 3 rows of 3 stickers in A 24 each row. If Mario’s sticker books have 4 rows of 3 B 8 B 6 stickers in each row, who has more stickers? Explain. C 5 × × = × × = > C 8 Tamisha has 3 (3 3) 27; Mario has 2 (4 3) 24; 27 24, D 52 D 21 so Tamisha has more stickers.

Module 8 • Lesson 2 247 248 © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company

Homework and Practice Use the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson.

247-248 Module 8