Investigate Model with Arrays

LESSON 6.6 Progress to Algebra Investigate Model with Arrays FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards 3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Also 3.OA.A.2 MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.) MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. F C R Coherence: Standards Across the Grades Before Grade 3 After 2.OA.C.3 3.OA.A.3 4.OA.A.2 F C R Rigor: Level 1: Understand Concepts...Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency... Your Own, Practice and Homework Level 3: Applications...Think Smarter and Go Deeper About the Math Professional Development Learning Objective Model division by using arrays. Language Objective Student teams make a scrapbook of examples to show how you can use arrays to solve division problems. Materials MathBoard, square tiles F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 299J. Partitive Division To model 12 4 3, start with 12 tiles. Progress to Algebra Using Arrays to Model Division In this lesson, students represent both partitive and quotative division situations with arrays. Form 3 rows. About the Math For partitive division, students make an array by setting up a given number of groups Development (rows) with one tile in each Professional row, then fill the rows, one tile at a time, until all the tiles are used. Then they count how many tiles are in each row to find the quotient. For quotative division, students set up one group (row) with a given number of tiles, then continue to make rows of the same number of tiles until all the tiles are used. Then they count the number of rows to find the quotient. The goal is to introduce students to another way to think about partitive and quotative division and connect the model or situation to the related division symbolism. Professional Development Videos 333A Chapter 6 Continue placing 1 tile in each row until all the tiles are used. The number in each row, 4, is the quotient. 12 4 3 5 4 Quotative Division To model 12 4 3, start with 12 tiles. Form 1 row of 3 tiles. Continue making rows of 3 tiles until all the tiles are used. The number of rows, 4, is the quotient. 12 4 3 5 4

Daily Routines Common Core Problem of the Day 6.6 Mr. Williams drove 213 miles in March. He drove 456 miles in April. About how many total miles did Mr. Williams drive in March and April? Estimate to the nearest hundred. Vocabulary 700 miles Interactive Student Edition Multimedia Glossary e 1 ENGAGE with the Interactive Student Edition Essential Question How can you use arrays to solve division problems? Making Connections Invite students to tell you what they know about division. How have you divided numbers? by making equal groups How can you model division? by using models or diagrams, or by acting out the division What questions can you answer with division? how many groups, or how many in a group Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. What is Lucia watching? Beetles cross the road. How many beetles are there in total? 28 How many beetles are in each group? 7 What does Lucia want to know? how many groups of 7 will cross What operation will Lucia use to find the answer? division Literacy and Mathematics Choose one or more of the following activities. Ask students how Lucia knows to use division to answer the question. Ask what prompts Lucia to ask how many groups will cross the road. Lucia notices the beetles are crossing in groups of 7. Have students work with partners to create a dialogue from the scenario and problem, perhaps with one student acting out Lucia and the other asking Lucia questions about what is happening and what Lucia wants to find out. How can you use arrays to solve division problems? 333B

LESSON 6.6 2 EXPLORE Investigate MATHEMATICAL PRACTICES MP2 Reason abstractly and quantitatively. Work together with students to complete the steps of the activity. Be sure students make equal rows of 5. Draw Conclusions What number did you divide? Explain. 30; possible answer: I started with 30 tiles. What number did you divide by? Explain. 5; possible answer: I made rows of 5 tiles each. MP8 Look for and express regularity in repeated reasoning. Exercise 3 requires students to generalize from the 5 by 6 array how to find a new array with 6 tiles in each row. Use the same array to describe a strategy for finding the number of rows of 6 tiles that are in 30. Possible answers: I knew that 30 5 = 6, so 30 6 = 5. Count the columns in the first array; Think: What number times 6 equals 30? MP4 Model with mathematics. Kara started with 18 tiles. She made rows of 3. She made 6 rows. What division equation did Kara model? 18 3 = 6 ELL Strategy: Identify Relationships Have students count the total number of desks in the classroom. Tell them you want to divide the desks into 5, 6, or 7 equal rows (depending on the total number of desks). Have students use tiles to show how to rearrange the desks to divide them into the given number of rows. How many desks are in each row? What division equation does the model represent? Progress to Algebra 3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Model with Arrays Essential Question How can you use arrays to solve division problems? Model with Arrays Investigate Materials square tiles You can use arrays to model division and find equal groups. A. Count out 30 tiles. Make an array to find how many rows of 5 are in 30. B. Make a row of 5 tiles. C. Continue to make as many rows of 5 tiles as you can. How many rows of 5 did you make? 6 rows Draw Conclusions 1. Explain how you used the tiles to find the number of rows of 5 in 30. 2. What multiplication equation could you write for the array? Explain. 6 5 = 30; possible explanation: there are 6 rows of 5 tiles each. You can use arrays to model division. How many rows of 6 tiles each can you make with 24 tiles? Use square tiles to make an array. Solve. Step 1 Use 24 tiles. Step 2 Make as many rows of 6 as you can. 3 2 1 Reteach Array Puzzles Use the clues to help solve the puzzle. You can use tiles or draw the array on a separate sheet of paper. 1. I am an array made with 24 tiles. I have 8 tiles in each row. How many rows do I have? Operations and Algebraic Thinking 3.OA.A.3 Also 3.OA.A.2 MATHEMATICAL PRACTICES MP4, MP7, MP8 Possible explanation: I placed the 30 tiles in rows of 5 tiles. Then I counted the number of rows I made. There were 6 rows. So, there are 6 rows of 5 in 30. 6 groups of 5 are 30. 3. Tell how to use an array to find how many rows of 6 are in 30. Possible answer: make rows of 6 until all 30 tiles are used. There are 5 rows of 6 in 30. Reteach 6.6 Problem Type: Arrays Unknown Factor Enrich 6.6 Chapter 6 333 Enrich 2. I am an array with 4 equal rows. I have 16 tiles in all. How many tiles are in each of my rows? 3 rows 4 tiles Differentiated Instruction 3. I am a square-shaped array. I have 7 rows. How many tiles do I have in all? (Hint: A square has 4 sides of equal length.) 4. I am an array made with 24 tiles. My number of rows is 2 more than the number of tiles in each of my rows. How many rows do I have? 333 Chapter 6 You can make 4 rows of 6. So, there are 4 rows of 6 tiles in 24. Use square tiles to make an array. Solve. Check students arrays. 1. How many rows of 7 are in 28? 2. How many rows of 5 are in 15? 4 rows Make an array. Then write a division equation. 3 rows Check students arrays. 3. 18 tiles in 3 rows 4. 20 tiles in 4 rows 18 4 3 5 6 20 4 4 5 5 5. 14 tiles in 2 rows 6. 36 tiles in 4 rows 14 4 2 5 7 36 4 4 5 9 Chapter Resources 6-15 Reteach 49 tiles 6 rows 5. I am an array with 7 tiles in each row. My number of rows is 4 less than the number of tiles in each of my rows. How many tiles am I made with in all? 21 tiles 6. I am an array made with 40 tiles. I have an odd number of rows and an even number of tiles in each of my rows. The number of my rows plus the number of tiles in each of my rows equals 13. How many rows do I have? 5 rows 7. Write your own array puzzle. Include the answer. Possible answer: I am a square-shaped array. I have 5 rows. How many tiles do I have in all? 25 tiles Chapter Resources 6-16 Enrich

Make Connections You can write a division equation to show how many rows of 5 are in 30. Show the array you made in Investigate by completing the drawing below. Make Connections Help students connect the array to a division equation. Why is 30 the dividend? because that is the number of tiles being divided, or separated into equal groups What represents the quotient in this array? The number of rows is the quotient. 30 5 = There are _ 6 rows of 5 tiles in 30. So, 30 5 = _. 6 Try This! Count out 24 tiles. Make an array with the same number of tiles in 4 rows. Place 1 tile in each of the 4 rows. Then continue placing 1 tile in each row until you use all the tiles. Draw your array below. How many tiles are in each row? 6 tiles What division equation can you write for your array? 24 4 = 6 or 24 6 = 4 334 Advanced Learners Math Talk You can divide to find the number of equal rows or to find the number in each row. Problem Type: Arrays Unknown Factor Possible explanation: it helps me solve the division problem by showing how many tiles are in each row. MATHEMATICAL PRACTICES 4 Use Models How does making an array help you divide? Visual / Kinesthetic Individual Materials 1-Centimeter Grid Paper (see eteacher Resources) Write the following numbers on the board. Try This! After students find the quotient, have them share how they wrote the division equation that the array represents. Then check their quotients. To divide, you have used drawing equal groups or circling equal groups, repeated subtraction, counting back on a number line, and arrays. Which method do you think is the easiest? Explain. Answers will vary. Accept all reasonable explanations. Have several students explain their choices. Math Talk Use Math Talk to reinforce how to use an array to divide. What information do you need in order to use an array to help you divide? What information do you find out by making the array? the total number (dividend) and the number in each row or the number of rows (divisor); You find the number of rows or the number in each row (quotient). COMMON ERRORS Error Students may make an incorrect array for a division problem. Example How many rows of 3 are in 18? 18 20 24 30 Have students choose one of the numbers. They should draw as many arrays as possible to represent that number on the grid paper. Each array should have from 1 to 10 rows and columns. Then have students write a corresponding division equation for each array. Springboard to Learning Review with students that if the problem asks for rows of 3, you put that number in each row and count the number of rows to get the answer. If the problem gives the number of rows, you start by making that many rows with one tile in each row. You keep adding one tile to each row until all the tiles are used. Then you count the number of tiles in each row. 334

3 Share and Show EXPLAIN MATH Share and Show MATH BOARD Exercises 5 8 are examples of partitive division. Before students complete the page, ask a volunteer to explain how he or she will find the answer to Exercise 5. Separate 25 tiles into 5 groups by placing one tile in each of 5 rows. Place one tile at a time in each row until all tiles are used. Count the number of tiles in each row to find the quotient. Remind students to write the division equations their arrays represent. Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard. MATH BOARD Use square tiles to make an array. Solve. 1. How many rows of 3 are in 18? 3. How many rows of 7 are in 21? 3 rows 2. How many rows of 6 are in 12? 4. How many rows of 8 are in 32? 4 rows Make an array. Then write a division equation. Check students arrays. Possible equations are given. 5. 25 tiles in 5 rows 6. 14 tiles in 2 rows 7. 28 tiles in 4 rows 6 rows 2 rows Check students arrays. 25 5 = 5 14 2 = 7 8. 27 tiles in 9 rows 28 4 = 7 27 9 = 3 If Quick Check a student misses the checked exercises 3 2 1 Rt I Problem Solving Applications 9. SMARTER Tell how to use an array to find how many rows of 8 are in 40. Possible answer: make rows of 8 tiles until all 40 tiles are used. There are 5 rows of 8 in 40. Then Differentiate Instruction with Reteach 6.6 Personal Math Trainer 3.OA.A.3 Rtl Tier 1 Activity (online) 10. MATHEMATICAL PRACTICE 4 Model Mathematics Show two ways you could make an array with tiles for 18 6. Shade squares on the grid to record the arrays. Check students drawings. 4 ELABORATE Chapter 6 Lesson 6 335 Problem Solving Applications SMARTER For Exercise 9, students should describe how to make the array and what it would look like. Remind students to write the division equations their arrays represent. Math on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. MP4 Model with mathematics. Exercise 10 requires students to make two separate drawings to model the division. PROBLEM TYPE SITUATIONS Addition and Subtraction Put Together/Take Apart Total Unknown Exercise: 12 Multiplication and Division Equal Groups Number of Groups Unknown Exercise: 12 Equal Groups Group Size Unknown Exercise: 13 Arrays Unknown Factor Exercise: 11 335 Chapter 6

336 MATHEMATICAL PRACTICES Unlock the Problem 11. MATHEMATICAL PRACTICE ANALYZE 7 Look for Structure Thomas has 28 tomato seedlings to plant in his garden. He wants to plant 4 seedlings in each row. How many rows of tomato seedlings will Thomas plant? a. What do you need to find? how many rows of tomato seedlings Thomas will plant b. What operation could you use to solve the problem? Possible answer: division c. Draw an array to find the number of rows of tomato seedlings. Check students drawings. There should be an array with 7 rows of 4 tiles. d. What is another way you could have solved the problem? Possible answer: I could have used repeated subtraction. DIFFERENTIATED INSTRUCTION e. Complete the sentences. Thomas has 28 tomato seedlings. He wants to plant 4 seedlings in each row. 12. DEEPER There were 20 plants sold at a store on Saturday, and 30 plants sold at the store on Sunday. Customers bought 5 plants each. How many customers in all bought the plants? 13. SMARTER Paige walked her dog 15 times in 5 days. She walked him the same number of times each day. How many times did Paige walk her dog each day? Shade squares to make an array to model the problem. _ 3 times 10 customers Differentiated Centers Kit So, Thomas will plant 7 rows of tomato seedlings. INDEPENDENT ACTIVITIES Unlock the Problem 5 MATHEMATICAL PRACTICES MP7 Look for and make use of structure. In Exercise 11, students draw an array to solve a problem in context. In Step d, have students share other strategies they could use to solve the problem. DEEPER Exercise 12 is a multistep problem involving addition and division. To extend the problem, have students write a number sentence that represents the problem. (20 + 30) 5 = 10 or (20 5) + (30 5) = 10 SMARTER Students must use an array to model the division situation given in the word problem. Some students will use mental math and know that they should draw an array that is 3 by 5 squares. Suggest to other students that they might try drawing rows of 5 and counting total squares after each row until they reach 15. Then they only need to count the rows to find the solution. EVALUATE Formative Assessment Essential Question Using the Language Objective Reflect Have student teams make a scrapbook of examples to answer the Essential Question. How can you use arrays to solve division problems? Possible answer: I can find how many equal groups by placing that number of tiles in each row of an array until all tiles are used. The number of rows is the answer. I can divide the tiles into a number of rows, placing 1 tile at a time in each row, until all the tiles are used. The number of tiles in each row is the answer. Activities Missing Sides Students complete blue Activity Card 19 by using division to find the length of a missing side. Literature The Garden Fence Students read the book and use division facts to find how much wood they need to build a fence. Math Journal WRITE Math Draw an array to show how to arrange 20 chairs into 5 equal rows. Explain what each part of the array represents. 336

Practice and Homework 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 Solving. Use the Write Math section to determine student s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. Model with Arrays Use square tiles to make an array. Solve. 1. How many rows of 4 are in 12? 3 rows Make an array. Then write a division equation. 3. 20 tiles in 5 rows Problem Solving 2. How many rows of 3 are in 21? 7 rows 4. 28 tiles in 7 rows 20 5 = 4 28 7 = 4 Practice and Homework COMMON CORE STANDARD 3.OA.A.3 Represent and solve problems involving multiplication and division. Check students arrays. 5. A dressmaker has 24 buttons. He needs 3 buttons to make one dress. How many dresses can he make with 24 buttons? 6. Liana buys 36 party favors for her 9 guests. She gives an equal number of favors to each guest. How many party favors does each guest get? 8 dresses 4 party favors 7. WRITE Math Draw an array to show how to arrange 20 chairs into 5 equal rows. Explain what each part of the array represents. Check students work. Chapter 6 337 Extend the Math Activity Area Models and Division Materials square tiles, 1-Centimeter Grid Paper (see eteacher Resources) Investigate Have students work with a partner to explore how to use an area model to solve a division problem. Have partners count out 20 tiles. Ask them to find how many rows with 5 tiles each they can make with the 20 tiles. Have students use the tiles to make an array. Then ask students to represent the array they made on grid paper. Have students share their strategies for making the area model on the grid. Then, ask students to draw a model on grid paper to show 2 equal rows with a total of 18 squares. Have them find how many squares are in each row. 9 squares Summarize Ask students to explain how drawing a model on grid paper to solve a division problem is like making an array and how it is different. Possible explanation: the grid and the array both show rows and columns. In the grid, there is no space between the rows and columns. Instead of placing tiles in an array, you color in squares on the grid paper. 337 Chapter 6

Lesson Check (3.OA.A.3) 1. Mr. Canton places 24 desks in 6 equal rows. How many desks are in each row? 4 desks 2. Which division equation is shown by the array? Possible answers: 12 3 = 4 or 12 4 = 3 Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. Spiral Review (3.OA.A.1, 3.OA.A.4, 3.OA.B.5, 3.OA.C.7) 3. Amy has 2 rows of 4 sports trophies on each of her 3 shelves. How many sports trophies does Amy have? 4. What is the unknown factor? 9 p = 45 24 sports trophies 5 5. Sam has 7 stacks with 4 quarters each. How many quarters does Sam have? 338 28 quarters 6. Skip count. How many counters are there in all? 18 counters FOR MORE PRACTICE GO TO THE Personal Math Trainer 338