2. Math Module 3 Multi-Digit Multiplication and Division Topic E: Division of Tens and Ones with Successive Remainders Lesson 15: Understand and solve division problems with a remainder using the array and area models 4.OA.3 4.NBT.6 PowerPoint designed by Beth Wagenaar Material on which this PowerPoint is based is the Intellectual Property of Engage NY and can be found free of charge at www.engageny.orgwww.engageny.org

3. Lesson 15 Target You will understand and solve division problems with a remainder using the array and area models.

4. Fluency Show Values with Number Disks Lesson 15 On your boards, write the number in standard form. Did you write 24? ThousandsHundredsTensOnes 

5. Fluency Show Values with Number Disks Lesson 15 On your boards, write the number in standard form. Did you write 53? ThousandsHundredsTensOnes 

6. Fluency Show Values with Number Disks Lesson 15 On your boards, write the number in standard form. Did you write 41? ThousandsHundredsTensOnes 

7. Fluency Show Values with Number Disks Lesson 15 On your boards, write the number in standard form. Did you write 41? ThousandsHundredsTensOnes   

8. Fluency Show Values with Number Disks Lesson 15 On your boards, write the number in standard form. Did you write 47? ThousandsHundredsTensOnes    

9. Fluency Show Values with Number Disks Lesson 15 32 Show 32 using number disks. ThousandsHundredsTensOnes 

10. Fluency Show Values with Number Disks Lesson 15 21 Show 21 using number disks. ThousandsHundredsTensOnes 

11. Fluency Show Values with Number Disks Lesson 15 43 Show 43 using number disks. ThousandsHundredsTensOnes 

12. Lesson 15 Fluency Divide with Remainders 2 How many groups of 2 are in 6? Let’s prove it by counting by twos. Use your fingers as you count. 4 6 Show and say how many groups of 2 are in 6.

13. Lesson 15 Fluency Divide with Remainders 2 How many groups of 2 are in 8? Let’s prove it by counting by twos. Use your fingers as you count. 4 6 8 Show and say how many groups of 2 are in 8.

14. Lesson 15 Fluency Divide with Remainders 3 How many groups of 3 are in 24? Let’s prove it by counting by three’s. Use your fingers as you count. 6 9 12 15 Show and say how many groups of 3 are in 24. 18 21 24

15. Lesson 15 Fluency Divide with Remainders 3 How many groups of 3 are in 25? Let’s prove it by counting by three’s. Use your fingers as you count. 6 9 12 15 How many groups? How many are left? 18 21 24

16. Lesson 14 Fluency Divide with Remainders 5 How many groups of 5 are in 15? Let’s prove it by counting by fives. Use your fingers as you count. 10 15 Show and say how many groups of 5 are in 15.

17. Lesson 15 Fluency Divide with Remainders 4 How many groups of 4 are in 12? Let’s prove it by counting by four’s. Use your fingers as you count. 8 12 Show and say how many groups of 4 are in 12.

18. Lesson 15 Fluency Divide with Remainders 8 How many groups of 8 are in 16? Let’s prove it by counting by eights. Use your fingers as you count. 16 How many groups of 8 are in 16?

19. Lesson 15 Fluency Divide with Remainders 8 How many groups of 8 are in 21? Let’s prove it by counting by eights. Use your fingers as you count. 16 How many groups? How many left?

20. Lesson 15 Fluency Divide with Remainders 9 How many groups of 9 are in 45? Let’s prove it by counting by nines. Use your fingers as you count. 18 27 36 45 How many groups of 9 are in 45?

21. Lesson 15 Fluency Divide with Remainders 9 How many groups of 9 are in 49? Let’s prove it by counting by nines. Use your fingers as you count. 18 27 36 45 How many groups? How many are left?

22. Lesson 15 Fluency Group Count to divide 2 468 6 1012

23. Lesson 15 Fluency Group Count to divide 2 468 7 1012 14

24. Lesson 14 Fluency Group Count to divide 5 10 15 20 8 2530 3540

25. Lesson 14 Fluency Group Count to divide 3 6 9 12 5 15

26. Lesson 15 Fluency Group Count to divide 5 10 15 20 6 2530

27. Lesson 15 Fluency Group Count to divide 5 10 15 20 5 25

28. Lesson 15 Fluency Group Count to divide 5 1

29. Lesson 14 Fluency Group Count to divide 3 6 9 12 7 1518 21

30. Lesson 15 Fluency Group Count to divide 3 6 9 12 5 15

31. Lesson 14 Fluency Group Count to divide 3 6 9 3

32. Lesson 15 Fluency Group Count to divide 48 12 16 9 2024 2832 36

33. Lesson 15 Fluency Group Count to divide 48 12 16 6 2024

34. Lesson 15 Fluency Group Count to divide 48 12 16 5 20

35. Lesson 15 Fluency Group Count to divide 48 12 16 4

36. Lesson 14 Fluency Group Count to divide 5 10 15 20 7 2530 35

37. Lesson 15 Fluency Group Count to divide 5 10 15 20 6 2530

38. Lesson 15 Fluency Number Sentences in an Array How many boxes do you see altogether? Let’s count by threes to check. 3 6 9 12 Let’s count by fives to check. 5 1010 15+115+1 On your boards, write two multiplication sentences to show how many boxes are in this array. 3 x 5 + 1 = 16 and 5 x 3 + 1 = 16 Write two division sentences. 15 + 1

39. Lesson 15 Fluency Number Sentences in an Array How many boxes do you see altogether? Let’s count by threes to check. 3 6 9 12 Let’s count by sixes to check. 6 1212 18+118+1 On your boards, write two multiplication sentences to show how many boxes are in this array. 3 x 6 + 1 = 19 and 6 x 3 + 1 = 19 Write two division sentences. 15 + 1 18

40. Lesson 15 Fluency Number Sentences in an Array How many boxes do you see altogether? Let’s count by threes to check. 3 6 9 12 Let’s count by fours to check. 4 8+18+1 12+112+1 On your boards, write two multiplication sentences to show how many boxes are in this array. 3 x 4 + 2 = 14 and 4 x 3 + 2 = 14 Write two division sentences. + 2

41. Application Problem 5 Minutes Lesson 15 Chandra printed 38 photos to put into her scrapbook. If she can fit 4 photos on each page, how many pages will she use for her photos?

42. Problem 1 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set Draw an array to represent 10 ÷2. Explain to your partner how you solved. I noticed some of you drew 2 circles and placed 10 dots evenly among the circles. 10 ÷ 2  Others of you drew 10 dots as 2 rows of 5 dots. 10 ÷2 = 5

43. Problem 1 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 10 ÷ 2 Let’s use graph paper to draw a rectangle with the area of 10 square centimeters and one side length of 2 centimeters. Tell your partner how we can find the unknown side length. 10 ÷2 Q = 5 The total is 10, so we know it is 5. If the width is 2 centimeters, that means the length is 5 centimeters, and 2 centimeters times 5 centimeters gives an area of 10 square centimeters. 10 2 5

44. Problem 1 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 10 ÷ 2 Q = 5 We can count and mark off by twos until we get to 10. X X X X X Discuss with your partner how the length of 5 centimeters is represented in the area model. The length is 5, and the quotient is 5. The length of the area model represents the quotient of this division problem.

45. Problem 1b Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 11 ÷ 2 X X X X X ? I can’t just draw 2 rows of square units because of the remainder. If I mark off 2 squares at a time, I count 2, 4, 6, 8, 10. I can’t do another group of 2 because it would be 12. There aren’t enough. With your partner, discuss how you would draw an area model for 11 ÷ 2. Two can be the length or the width.

46. Problem 1b Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 11 ÷ 2 Eleven square centimeters is the total area. Let’s draw a rectangle starting with a width of 2 centimeters. We’ll continue lengthening it until we get as close to 11 square centimeters as we can. A length of 5 centimeters and width of 2 centimeters is as close as we can get to 11 square centimeters. We can’t do 2 × 6 because that’s 12 square centimeters and the total area is 11 square centimeters. We can show a total area of 11 square centimeters by modeling 1 more square centimeter. The remainder of 1 represents 1 more square centimeter. 11 ÷2 Q = 5 R = 1 10 5 2

47. Problem 1c Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 16 ÷ 3 X X X X X X X X X X ? I can’t just draw 3 rows of square units because of the remainder. If I mark off 3 squares at a time, I count 3, 6, 9, 12, 15. I can’t do another group of 3 because it would be 18. There aren’t enough. With your partner, discuss how you would draw an area model for 16 ÷ 3. Three can be the length or the width.

48. Problem 1c Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 16 ÷ 3 Sixteen square centimeters is the total area. Let’s draw a rectangle starting with a width of 3 centimeters. We’ll continue lengthening it until we get as close to 16 square centimeters as we can. A length of 5 centimeters and width of 3 centimeters is as close as we can get to 16 square centimeters. We can’t do 3 × 6 because that’s 18 square centimeters and the total area is 16 square centimeters. We can show a total area of 16 square centimeters by modeling 1 more square centimeter. The remainder of 1 represents 1 more square centimeter. 16 ÷3 Q = 5 R = 1 15 5 3

49. Problem 1d Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 23 ÷ 4 X X X X X X X X X X ?????? I can’t just draw 4 rows of square units because of the remainder. If I mark off 4 squares at a time, I count 4, 8, 12, 16, 20. I can’t do another group of 4 because it would be 24. There aren’t enough. With your partner, discuss how you would draw an area model for 23 ÷ 4. Four can be the length or the width.

50. Problem 1d Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 23 ÷ 4 Twenty three square centimeters is the total area. Let’s draw a rectangle starting with a width of 4 centimeters. We’ll continue lengthening it until we get as close to 23 square centimeters as we can. A length of 5 centimeters and width of 4 centimeters is as close as we can get to 23 square centimeters. We can’t do 4 × 6 because that’s 24 square centimeters and the total area is 23 square centimeters. We can show a total area of 23 square centimeters by modeling 3 more square centimeters. The remainder of 3 represents 3 more square centimeter. 23 ÷4 Q = 5 R = 3 20 5 4

51. Problem 2 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 38 ÷ 4 In the Application Problem, you drew an array (pictured to the right) to solve. Represent the same problem using the area model on graph paper. You will have two minutes to work. What do you notice about the array compared to the area model on graph paper?

52. Problem 2 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 38 ÷ 4 The area model is faster to draw. Thirty-eight dots is a lot to draw. There are the same number of dots and squares when we used graph paper. We get the same answer of a quotient 9 with a remainder of 2. Let’s represent 38 ÷ 4 even more efficiently without graph paper since it’s hard to come by graph paper every time you want to solve a problem. You have one minute to draw.

53. Problem 2 Solve a division problem with and without a remainder using the area model. Lesson 15 Problem Set 38 ÷ 4 Talk to your partner about how the array and graph paper models supported you in drawing the rectangle with a given structure. I knew the length was a little more than twice the width.  I knew that the remainder was half a column. I knew that there was a remainder. It was really obvious with the array and graph paper.

54. Problem Set 10 Minutes

55. Problem Set 10 Minutes Explain to your partner how Problem 1(a) and Problem 1(b) are similar. How are they different?

56. Problem Set 10 Minutes

57. Problem Set 10 Minutes How can Problem 3 and Problem 4 have the same remainder?

58. Problem Set 10 Minutes How could you change the 43 in Problem 5 so that there would be the same quotient but with no remainder?

59. What does the quotient represent in the area model? When does the area model present a challenge in representing division problems? The quotient represents a side length. The remainder consists of square units. Why? How is the whole represented in an area model? What new math vocabulary did we use today to communicate precisely? How did the Application Problem connect to today’s lesson? Debrief Lesson Objective: You will understand and solve division problems with a remainder using the array and area models.

60. Exit Ticket Lesson 1