Algebra • Distributive Property

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LESSON 4.4 Algebra • Distributive Property

FOCUS COHERENCE RIGOR LESSON AT A GLANCE

F C R Focus: Common Core State Standards Learning Objective 3.OA.B.5 Apply properties of operations as strategies to multiply Use the Distributive Property to ﬁnd products by and divide. breaking apart arrays. Also 3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.C.7 MATHEMATICAL PRACTICES Language Objective MP4 Model with mathematics. MP7 Look for and make use of structure. Students give two ways you can use the MP8 Look for and express regularity in repeated reasoning. Distributive Property to ﬁnd products.

F C R Coherence: Materials Standards Across the Grades MathBoard, square tiles Before Grade 3 After 2.OA.A.1 3.OA.B.5 4.OA.A.2 F C R Rigor: Level 1: Understand Concepts...... Share and Show ( Checked Items) F C R For more about how GO Math! fosters Coherence Level 2: Procedural Skills and Fluency...... On Your Own within the Content Standards and Mathematical Progressions Level 3: Applications...... Think Smarter and Go Deeper for this chapter, see page 189J.

About the Math Professional Development

Model the Distributive Property 8 3 7 Models help students develop a better understanding of concepts. StudentsAbout represent the the Math concepts ﬁrst with concrete materialsProfessional or drawings, Development then with numbers and symbols. A model also helps students visualize the process and form a mental image. In this lesson, a factor is partitioned using an array to show why the Distributive Property works. The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. 52 The goal of this lesson is to use the Distributive Property as a strategy to partition one of the factors. In this way, facts with greater products are shown as the sum of the products of two facts that students already know, making 8 × (5 + 2) multiplication faster and more ﬂexible. (8 × 5) + (8 × 2) 40 + 16 = 56 Professional Development Videos

209A Chapter 4 1 ENGAGE

Daily Routines with the Interactive Student Edition Common Core Essential Question Problem of the Day 4.4 How can you use the Distributive Property to find products? Ellis has 6 boxes. Each box holds 1 pencil. Write a number sentence to show how Making Connections many pencils Ellis has in all. Invite students to tell you what they know about multiplying by 1, 5, and 6.

6 × 1 = 6 What happens when you multiply any number by 1? The answer is the same number.

Vocabulary Distributive Property What are different ways to multiply by 5? count by 5, add 5s What are different ways to multiply by 6? count by 6, add 6s Interactive Student Edition Multimedia e Glossary Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. • How far does Lucia’s neighbor drive each day for work? 7 miles • How many days per week does Lucia’s neighbor drive to and from work? 5 days • What operation could you use to find the total number of miles Lucia’s neighbor drives for work per week? multiplication or addition • What multiplication expression could you use to find the total number of miles Lucia’s neighbors drives? 5 × 7 • What are different ways to solve 5 × 7? count by 7s, add 7 + 7 + 7 + 7 + 7, multiply 5 × 6 and add another 5 Literacy and Mathematics • Have students discuss the meaning of distribute in a larger context. Ask one student to distribute pieces of paper to each student to demonstrate the meaning of the word. • Have students write a short story about why Lucia wants to know how many miles his neighbor drives.

How can you use the Distributive Property to find products?

Lesson 4.4 209B LESSON 4.4 3.OA.B.5 Apply properties of operations as strategies to multiply and divide. 2 EXPLORE

ALGEBRA Name Lesson 4.4 Unlock the Problem Distributive Property MATHEMATICAL PRACTICES Essential Question How can you use the Distributive Property to Operations and Algebraic find products? Thinking—3.OA.B.5 Also 3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.C.7 MATHEMATICAL PRACTICES Students should understand that they need MP1, MP7, MP8 to multiply the number of fish by the cost of UnlockUnlock the ProblemProblem each fish to find the total amount spent. Mark bought 6 new fish for his • Describe the groups in Have a volunteer read the definition of the aquarium. He paid $7 for each fish. this problem. How much money did he spend in all? Distributive Property. 6 groups of $7

Hands On Find 6 × $7. • Circle the numbers you will use to Activity solve the problem. You can use the Distributive Property MP4 Model with mathematics. Work with to solve the problem. students to make the array and break it apart. • How can you use the smaller arrays to ﬁnd The Distributive Property states that multiplying a sum by a number is the sum—the answer to an addition problem the total? Find the total for each smaller array and same as multiplying each addend by the addends—the numbers being added add them together. number and then adding the products.

Hands MP6 Attend to precision. On Materials ■ square tiles • Does breaking apart a larger array into Activity 2 smaller arrays make the problem easier Make an array with tiles to show Break apart the array to make two 6 rows of 7. smaller arrays for facts you know. to solve? Explain. Yes; Possible explanation: I can use basic facts I already know to ﬁnd the total for each smaller array and then add the totals to ﬁnd the answer. • Why do you think this is called the Distributive Property? Possible answer: because I am distributing 6 to both 2 and 5 to multiply. 6 × 7 = ■ 6 × 5 6 × 2 6 × 7 = ■ 6 × 7 = 6 × (5 + 2) Think: 7 = 5 + 2 Math 6 × 7 = (6 × 5) + (6 × 2) Multiply each addend by 6. Talk Use Math Talk to focus on 30 12 Math students’ understanding of using a 6 × 7 = _ + _ Add the products. Talk MATHEMATICAL PRACTICES 2 Reason Quantitatively model to draw conclusions and explain reasons 6 × 7 = _42 for their conclusions. Have several students What other ways could 42 you break apart the

© Houghton Mifflin Harcourt Publishing Company So, Mark spent $_ for his new fish. draw their arrays on the board and write the 6 × 7 array? Possible answers: 6 × (6 + 1); 6 × (4 + 3) multiplication problems they used to ﬁnd 6 × 7. Chapter 4 209 • Can you separate the array into 2 rows of 6 and 5 rows of 6 and get the same answer? Use the Distributive Property to see if you get the same answer. Possible 3 Differentiated answer: (2 × 6) + (5 × 6) = 12 + 30, or 42. Yes, it Reteach 4.4 2 Enrich 4.4 1 Instruction works; I get the same answer.

Lesson 4.4 Lesson 4.4 Name Reteach Name Enrich ELL Strategy: Algebra • Distributive Property United Arrays A garden has 4 rows of 7 corn stalks. How many corn stalks Draw an array for each clue. Then use the arrays Check students’ Rephrase in all are in the garden? to solve each problem. arrays.

You can use the Distributive Property to break an array 1. Craig spent $27 to buy 3 into smaller arrays to help you fnd the answer. calendars. The next day, he spent Explain the Distributive Property using an another $18 for more calendars. Find 4 3 7. Each calendar cost the same illustrated example from the lesson as support. amount. How many calendars did Step 1 Make an array to show 4 rows of 7. Craig buy? 5 calendars • In pairs, have students draw their own 4 rows of 7, or 4 3 7 Step 2 Break apart the array to make two 2. On Monday, Mrs. Jones spent array and break it apart, or distribute it, by smaller arrays for facts you know. $32 on 4 books. On Tuesday, she spent $16 on more books. Each book cost the same amount. How drawing the array and a dotted line where Step 3 Write the multiplication for the new many books did Mrs. Jones buy? arrays. Multiply and then add the products 4 3 4 1 4 3 3 they would break it apart. to fnd the answer. 16 1 12 5 28 6 books 4 3 7 5 (4 3 4) 1 (4 3 3) 4 3 7 5 16 1 12 4 3 7 5 28 3. Hailey spent $12 to buy 2 fish. Her cousin spent double the So, there are 28 corn stalks in all in the garden. • Have them write the multiplication amount on fish. Each fish cost the same amount. How many fish expression for the array and then for the Write one way to break apart the array. Possible answers are given. do Hailey and her cousin have Then fnd the product. altogether?

broken apart, or distributed, array. 1. 2. 6 f sh

4. Explain how you used the arrays to solve each problem. • Have the pairs express the property or rule Possible answer: I drew two arrays for each problem then added the number of rows to in their own words in their Math Journal. (3 3 4) 1 (3 3 5); or (6 3 1) 1 (6 3 2); or f nd the number of items. (1 3 9) 1 (2 3 9); 27 (3 3 3) 1 (3 3 3); 18

Chapter Resources 4-11 Reteach Chapter Resources 4-12 Enrich 209 Chapter 4 © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A CorrectionKey=ATry This! Have students draw other possible arrays for 9 × 6 and verify that they work. Try This! Check students’ drawings. Possible answers are given. (2 × 6) + (7 × 6) and (6 × 6) + (3 × 6). Suppose Mark bought 9 fish for $6 each. • Which array had the easiest numbers to You can break apart a 9 × 6 array into two smaller arrays for facts you know. One way is work with? Possible answers: (5 × 6) + (4 × 6) _56 × _ to think of 9 as 5 + 4. Draw a line to show this because it is easy to multiply by 5 and then double way. Then find the product. 2 × 6 and add the numbers. 30 + 24 = 54; 9 × 6 = (__5 × __646) + (__ × __) (6 × 6) + (3 × 6) because they are both facts I know. • Could you solve the problem another way 9 × 6 = __30 + __ 24 _4 × _6 using the Distributive Property? Yes; Possible So, Mark spent $_54 for 9 fish. answer: break apart the 6 and get (9 × 3) + (9 × 3).

Hands That would be 27 + 27 = 54. MATH On Share and Show BOARD

1. Draw a line to show how you could break apart this 6 × 8 array into two smaller arrays for 3 EXPLAIN facts you know. Possible answers are given.

64 • What numbers do you multiply? __ and __ Hands MATHMATH On 64 Share and Show BOARD __ and __ Possible answer: I broke the • What numbers do you add? __24 + __24 larger array into two smaller The ﬁrst problem connects to the learning arrays, so I have to add their 6 × 8 = 6 × (__4 + __4 ) products to f nd the total product. model. Have students use the MathBoard to explain their thinking. 6 × 8 = (__6 × __4 ) + (__64 × __) Math MATHEMATICAL PRACTICES 7 Use the checked excercises for Quick Check. 6 × 8 = __24 + __24 Talk Look for Structure Why Students should show their answers for the 6 × 8 = __48 do you have to add to find the total product Quick Check on the MathBoard. when you use the Write one way to break apart the array. Distributive Property? Then find the product. Possible answers are given for 2–3. 3 3 2 2 2. 3. Quick Check 1 RtIR Quick Check 1 RtI

If a student misses the checked excercises

5 × (4 + 4) or (2 + 3) × 8 4 × (4 + 3) or (2 + 2) × 7 Then Differentiate Instruction with

© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company • Reteach 4.4 or (2 × 8) + (3 × 8); 40 or (2 × 7) + (2 × 7); 28 • Personal Math Trainer 3.OA.B.5 210 • RtI Tier 1 Activity (online)

3_MNLESE342101_C04L04.indd 210 11/02/14 12:19 PM COMMON ERRORS AdvancedAdvanced Learners Learners Logical / Mathematical Individual ErrorCOMM When usingON the Distributive ERRORS Property, students may break a number into its factors • Write the following problem on the board. instead of its addends. Example 6 × 9 = (6 × 3) + (6 × 3) 9 × 6 = ? The student broke apart 9 into two factors, 3 and 3, instead of two • Challenge students to apply the Distributive addends, such as 3 and 6. Property, breaking apart one factor into 3 addends. Have them show each step of the Springboard to Learning Encourage process. Possible answers: students to write out their addition sentences (2 × 6) + (3 × 6) + (4 × 6) = 12 + 18 + 24 = 54 before writing their multiplication sentences. (3 × 6) + (3 × 6) + (3 × 6) = 18 + 18 + 18 = 54 For example: 6 × 9 = 6 × (6 + 3) = (6 × 6) + (6 × 3). Also have students use • Then have students write another fact and repeat square tiles to reinforce the concept of breaking the process. apart a larger array into two smaller arrays.

Lesson 4.4 210 211 that theotherfactoralsocanbebrokenapart. one correctanswer,theymightnotrecognize Distributive Property. If the students select only can bebrokenapartintoaddendsusingthe Students mustrecognizethateitherfactor answer shouldbe. explanation shouldincludewhatthecorrect explain whyRobin’swayisnotcorrect.Their the DistributiveProperty.Thentheyshould an errorthroughtheirunderstandingof Students mustﬁrstrecognizethatthereis reasoning. statement toﬁndanerrorandjustifytheir Exercise 5requiresstudentstoanalyzea critique thereasoningofothers. MP3 Constructviableargumentsand Own section. correctly, theymaycontinuewiththeOnYour If studentscompletethecheckedexcercises On YourOwn product, andexplanation. that thearrayshadedmatchesfact, Exercise 4hasseveralpossibleanswers.Check DEEPER Chapter Student Editionandat Math ontheSpot SMARTER SMARTER solve thistypeof Use thisvideotohelpstudentsmodeland Video Tutor Math ontheSpot

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On YourOwn 9 7 think can I No; possible answer:No;possible can Robin 9 Possible explanation: I shaded 9 rows of 7 tiles.7 rowsof 9 Possible shaded explanation:I 7a. 7d. 7c. 7b. to indicate whether thesum orproduct to 7 isequal receive? He gives the clerk $40.How much change should Kent howthe fact. you Explain found theproduct. shows 7,8,or9asafactor. afactwith Write Justify your answer. 3 doubling it. If she it.If doubling × × ×

7 7 DEEPER DEEPER 7anddoubling it.“ herstatement Does make sense? 7 (5 (7 7 SMARTER SMARTER = = + × × × 9 36 (3 (3 4) 2) Kent buys 5bags ofpotatoes that cost $7each. Shade tilesto make anarray that × + + + (4 + + For numbers Yes 7a-7d,choose orNo Robin says, can “I find8 = 27 or 63 or 27 2) 2) (5 (7 4 + = 3);9 = + × × f nds 3 nds

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What’s the Error? 4 ELABORATE

MATHEMATICAL 8. PRACTICE 3 Verify the Reasoning of Others Brandon needs 8 boxes of spinners for his fishing club. The cost of each box is $9. Problem Solving • Applications How much will Brandon pay? MATHEMATICAL PRACTICES 8 × $9 = ■

Look at how Brandon solved the problem. Find and describe his error. What's the Error? MP3 Construct viable arguments and 8 × 9 = (4 × 9) + (5 × 9) critique the reasoning of others. 8 × 9 = 36 + 45 For Exercise 8, have students read the problem and discuss what they need to find. 8 9 81 × = This problem requires students to find and describe an error. Then they need to find the correct answer, showing each step. If students have trouble finding the error Possible explanation: Brandon broke apart the factor 9 and multiplied by 9. that Brandon made, have students solve the He should have broken apart the factor 8 or multiplied by 8. problem and then compare their answer to his to find the error.

Use the array to help solve the problem Some students might realize that and correct his error. Brandon could have used the Distributive Property to solve (4 × 8) + (5 × 8) to get 8 × 9 = (4 + 4) × 9 the correct answer of 72. Have students 8 × 9 = (_4 × _94) + (_ × _9 ) share any possible strategies they found to solve the problem. 8 × 9 = _36 + _ 36

8 × 9 = _72 Formative 5 EVALUATE Assessment

212 Essential Question Using the Language Objective Reflect Have students answer the Essential Question. How can you use the Distributive Property DIFFERENTIATEDD INSTRUCTION INDEPENDENT ACTIVITIES to find products? Possible answer: I can break apart one of the factors and multiply each addend by the other factor. Then I can add the products to find the answer.

Differentiated Centers Kit Math Journal WRITE Math What are some ways you could break apart Activities Activities Literature 7 × 9 using the Distributive Property? Hurray for Arrays! Factor Spin Collections Times Four

Students complete Students complete Students read blue Activity purple Activity the book and Card 15 by using Card 7 by using determine how to arrays to model randomly generated use multiplication multiplication facts. factors to practice to ﬁnd the total multiplication facts number of objects for 1, 2, 4, and 5. in each collection. Lesson 4.4 212 Practice and Homework Name Lesson 4.4 Distributive Property Possible answers are given. COMMON CORE STANDARD—3.OA.B.5 Understand properties of multiplication and Write one way to break apart the array. the relationship between multiplication and Then find the product. division. Practice and Homework 1. 2. Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem (3 × 7) + (3 × 7) Solving. Use the Write Math section to ______(7______× 4) + (7 × 4) or determine student’s understanding of content ______42 (7______× 5) + (7 × 3); 56 for this lesson. Encourage students to use their Math Journals to record their answers.

ProblemProblem Solving

3. There are 2 rows of 8 chairs set up 4. A marching band has 4 rows of in the library for a puppet show. trumpeters with 10 trumpeters in each How many chairs are set up? Use the row. How many trumpeters are in the Distributive Property to solve. marching band? Use the Distributive Property to solve.

______16 chairs ______40 trumpeters

5. WRITE Math What are some ways you could break apart 7 × 9 using the Distributive Property? Check students’ work.

213 Chapter 4 TEST PREP Lesson Check (3.OA.B.5) 1. Complete the number sentence to 2. What is one way to break apart show the Distributive Property. the array? Continue concepts and skills practice with 7 × 6 = Lesson Check. Use Spiral Review to engage students in previously taught concepts and Possible answers: (7 × 3) + (7 × 3); to promote content retention. Common Core standards are correlated to each section. ______(7 × 2) + (7 × 4); (7 × 1) + (7 × 5); Possible______answers: (2 × 6) + (2 × 6)

______(3 × 6) + (4 × 6), (2 × 6) + (5 × 6); or______(4 × 3) + (4 × 3)

______(1 × 6) + (6 × 6) ______

Spiral Review (3.NBT.A.1, 3.NBT.A.2, 3.MD.B.3) 3. The school auditorium has 448 4. Find the difference. chairs set out for the third-grade performance. What is 448 rounded 400 to the nearest ten? __ − 296 104

______450

5. There are 622 fruit snacks in one crate 6. Which sport do exactly 6 students and 186 in another crate. How many play? fruit snacks are there? Sports Students Play

Football 622 Baseball + 186 __ Sport Basketball

Soccer

0 2 4 6 8 10 12 14 16 Number of Students

______808 fruit snacks ______Football © Houghton Mifflin © HarcourtCompany Houghton Mifflin Publishing FOR MORE PRACTICE GO TO THE 214 Personal Math Trainer

Lesson 4.4 214