1. This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

2. 4th Grade Multiplication & Division Relationship www.njctl.org 2012-07-18

3. Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 7 for an example.) Setting the PowerPoint View

4. Table of Contents Click on a topic to go to that section. Multiplication Review Factors and Multiples Prime and Composite Inverse Operations

5. Multiplication Review Return to table of contents

6. The answer to a multiplication problem is called the product. The numbers you multiply are called factors. Here are two ways to write multiplication facts: factor product factor product Terms to remember: 3(5) = 15 3 X 5 = 15

7. 1 Using the multiplication sentence, 6 x 8 = 48 Which number is the product?

8. 2 Using the multiplication number sentence, 9 x 5 = 45 Which number is a factor? A 9 B 5 C 45 D 0

9. 3What is the product for 6 x 6?

10. 4What is the product for 7(9)?

11. Multiplication is a fast way of adding a series of numbers 5 X 3 means 5 + 5 + 5 5 x 3 = 15 Click for answer

12. The commutative property of multiplication means It does not matter which number is first when you write the problem 3 X 5 is the same as 5 X 3 (They both equal 15) or a x b = b x a 3 x 8 = 8 x 3 24 = 24

13. Any number multiplied by 0 is always zero 0 X 3 = 0 + 0 + 0 = 0

14. Any number times ONE is always itself 5 x 1 = 5

15. 5 425 x 0 =

16. 6 425 x 1 =

17. 7If 8 x 3 = 24, what does 3 x 8 = ?

18. 8 0 x 301 =

19. 9 1 x 301 =

20. 10 There are four birthday cakes with seven candles on each cake. How many candles are there in all?

21. 11What would the multiplication number sentence look like for the repeated addition problem, 7 + 7 + 7 + 7? A B C D 4 x 4 4 x 7 7 x 7 7 + 4

22. 12Which property is shown? 5 x 4 = 20 4 x 5 = 20 A B C D Same Zero Commutative Identity

23. 13 Which set of number sentences show the commutative property? A B C D 3 + 3 = 6 2 + 2 + 2 = 6 8 x 2 = 16 2 x 8 = 16 4 x 1 = 4 0 x 4 = 0 7 x 3 = 21 7 + 7 + 7 = 21

24. Multiplying Using a Model Find 3 x 5 Arrange 5 rows of 3 Using stars to represent ones, you count 15 stars. 15 is a multiple of 3.

25. rows When writing a multiplication number sentence for an array, write the number of rows first. The second number should be the number in each row. 4 x 2 2 x 4 columns

26. Drag arrows into the rectangle to make an array showing 4 x 6

27. 5 x 2 2 x 5 Drag arrows into each rectangle to make the arrays. How are they the same? Different?

28. On dot paper, draw several arrays. For example: 8 x 3

31. 14 Which array is a model for 3 x 4? A B C Dnone of the above

32. 15 This is array shows: A B C D 0 x 3 3 x 0 3 x 1 1 x 3

33. 16 Which array is circled? A B C D 10 x 7 8 x 5 2 x 4 x 5 5 x 8

34. Prime and Composite Return to table of contents

35. Prime Numbers have only 2 factors 1 and itself Example 5 = 1 x 5 Composite Numbers have more than 2 factors Example 30 = 5 x 6

36. Sort the numbers into the columns. Prime Composite 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0

37. Special Cases: 0 and 1 - neither prime nor composite

38. Label the numbers as either Prime or Composite 22 23 24 27 29 31 33 35 37 39 41 43 44 47 51 50

39. 111 12 59 54 62 20 78 132 98 18 23 126 81 76 9 54 42 45 41 47 5 139 108 53 109 112 72 83 20 103 14 44 97 29 126 3 36 102 98 123 41 130 138 61 121 98 123 134 127 50 82 11 17 2 97 19 37 110 Circle the prime numbers to help the space shuttle to take off from Earth

40. 17 Which of the following numbers are prime? (Select more than one answer.) A B C D 4 3 2 1 E F 5 6

41. 18 Which of the following numbers are composite? (Select more than one answer.) A B C D 12 11 10 9 E F 13 14

42. 19Which of the following sets of numbers has all prime numbers? A B C D 2, 3, 5, 7, 11 0, 1, 2, 3, 5 2, 3, 5, 7, 9 1, 2, 3, 5, 7

43. 20Which of the following sets of numbers has all composite numbers? A B C D 12, 14, 15, 16, 17 2, 6, 9, 11, 15 4, 6, 8, 9, 10 2, 4, 6, 8, 10

44. Hundreds Chart Activity: By crossing out multiples of numbers, all of the prime numbers will be identified. Use red to cross out all of the even numbers (2, 4, 6, etc.) Use blue to cross out all of the multiples of 3 (3, 6, 9, etc.) that remain. Use purple to cross out the multiples of 5 that remain.

46. Click for answer

47. Click for game.

48. Factors and Multiples Return to table of contents

49. Factors are the numbers you multiply to get another number. For example, the factors of 35 are 5 and 7, because 5 x 7 = 35. Some numbers have more than two factors. For example, 30 has 8 factors 1 x 30 and 5 x 6 and 2 x 15 and 3 x 10

50. 21 Select all of the factors for the number 27. A B C D 9 5 3 1 E F 14 27

51. 22 Select all of the factors for the number 5. A B C D 4 3 2 1 E 5

52. 23 Select all of the factors for the number 20. A B C D 4 and 5 3 and 15 2 and 10 1 and 10 E F 4 and 6 6 and 12

53. It is easier to find of the factors, if they are organized into a factor rainbow. Factors of 20: 1, 2, 4, 5, 10, 20 Click

54. Steps to find all of the factors of a number. What are all of the factors of 42? Step 1: Always start with 1 times the number. 1 42 Step 2: If it is even then divide it by 2 to get the other factor. Step 3: Check if it is divisible by 3. Step 4: Check other numbers (4,5,6, etc.) Step 5: Make the factor rainbow.

55. Steps to find all of the factors of a number. What are all of the factors of 42? Step 1: Always start with 1 times the number. 1 42 Step 2: If it is even then divide it by 2 to get the other factor. 1, 2 21, 42 Step 3: Check if it is divisible by 3. Step 4: Check other numbers (4,5,6, etc.) Step 5: Make the factor rainbow.

56. Steps to find all of the factors of a number. What are all of the factors of 42? Step 1: Always start with 1 times the number. 1 42 Step 2: If it is even then divide it by 2 to get the other factor. 1, 2 21, 42 Step 3: Check if it is divisible by 3. 1, 2, 314, 21, 42 Step 4: Check other numbers (4,5,6, etc.) Step 5: Make the factor rainbow.

57. Steps to find all of the factors of a number. What are all of the factors of 42? Step 1: Always start with 1 times the number. 1 42 Step 2: If it is even then divide it by 2 to get the other factor. 1, 2 21, 42 Step 3: Check if it is divisible by 3. 1, 2, 314, 21, 42 Step 4: Check other numbers (4,5,6, etc.) No others in between 3 and 14. Step 5: Make the factor rainbow.

58. Steps to find all of the factors of a number. What are all of the factors of 42? Step 5: Make the factor rainbow. 1, 2, 3, 14, 21, 42 The factors of 42 are: 1, 2, 3, 14, 21, 42

59. 24 Which correctly lists all of the factors for 40 ? A B C D 1, 2, 20, 40 1, 2, 4, 5, 8, 10, 20, 40 1, 2, 3, 4, 5, 8, 9, 10, 20, 40 1, 40

60. 25 Which correctly lists all of the factors for 31 ? A B C D 1, 3, 7, 9, 31 1, 3, 9, 31 1, 3, 31 1, 31

61. 26 Which correctly lists all of the factors for 24 ? A B C D 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 3, 4, 6 1, 2, 4, 6, 12, 24 1, 24

62. Being able to factor numbers is a key skill, one that is necessary to learn and perform many other math skills down the line. Instead of just finding all of the factors, we can find the prime factors. A prime factor a number that can only be divided by itself and 1.

63. is the process of factoring a number so that all of the factors are prime numbers. Prime Factorization

64. Three different methods are shown. All lead to the same result. Method 1: Method 2: Method 3: Factor Tree Columns Stairs 2 40 2 20 2 10 5 5 1 2 x 2 x 2 x 5 = 2 3 x 5 2 40 1 5 5 2 10 2 20 2 40 4 10 2 5 2

65. Prime Factorization Method 1: Create a factor tree. The first branch consists of any two facts of the composite number. Add branches to your factor tree until you only have prime factors.

66. 48 12 4 4 3 2 2 22 2 x 2 x 3 x 2 x 2 = 2 4 x 3

67. STEPS For Method 2 Prime Factorization of 60 1. Write the number and make columns as shown. 2. Start with smallest prime number that is a factor of the number. Write that to the left of the number. 3. Divide the number by what you wrote on the left. Write the answer on the right. 4. Continue this division by primes until you are left with one on the right. 5. The numbers on the left are the prime factors. 60

68. STEPS For Method 2 Prime Factorization of 60 1. Write the number and make columns as shown. 2. Start with smallest prime number that is a factor of the number. Write that to the left of the number. 3. Divide the number by what you wrote on the left. Write the answer on the right. 4. Continue this division by primes until you are left with one on the right. 5. The numbers on the left are the prime factors. 2 60

69. STEPS For Method 2 Prime Factorization of 60 1. Write the number and make columns as shown. 2. Start with smallest prime number that is a factor of the number. Write that to the left of the number. 3. Divide the number by what you wrote on the left. Write the answer on the right. 4. Continue this division by primes until you are left with one on the right. 5. The numbers on the left are the prime factors. 2 60 30

70. STEPS For Method 2 Prime Factorization of 60 1. Write the number and make columns as shown. 2. Start with smallest prime number that is a factor of the number. Write that to the left of the number. 3. Divide the number by what you wrote on the left. Write the answer on the right. 4. Continue this division by primes until you are left with one on the right. 5. The numbers on the left are the prime factors. 2 60 2 30 3 15 5 1 click 2 x 2 x 3 x 5 = 2 2 x 3 x 5 Click

71. STEPS For Method 3 Prime Factorization of 12 1. Divide the given number by the smallest prime number possible. Organize your work in steps. 2. Continue to divide by the smallest prime number possible. 3. Keep dividing until the quotient (answer) is one. 212 3 6 3 2 Example: 1 12 = 2 x 2 x 3 = 2 2 x 3

72. What is the prime factorization of 18? 2 3 3 18 9 3 1 18 = 2 x 3 x 3 = 2 x 3 2 click for answer

73. What is the prime factorization of 24? 2 3 24 12 6 3 2 2 1 24 = 2 x 2 x 2 x 3 = 2 3 x 3 click for answer

74. Click for interactive web site to find the prime factorization of 1 number.

75. 27What is the prime factorization of 30? A B C D 2 x 15 5 x 6 6 x 5 2 x 3 x 5

76. 28What is the prime factorization of 45? A B C D 5 2 x 3 9 x 5 3 2 x 5 3 x 15

77. 29What is the prime factorization of 100? A B C D 2 2 x 25 2 2 x 5 2 2 x 5 2 2 x 3 x 10

78. 30What is the prime factorization of 49? A B C D 7272 4 x 9 1 x 49 7

79. 31What is the prime factorization of 36? A B C D 2 x 3 2 2 x 9 2 x 3 3 2 2 x 3 2

80. Multiples of a number are the products of the number and other factors. Multiply to find multiples.

81. 1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 4 x 3 = 12 5 x 3 = 15 6 x 3 = 18 7 x 3 = 21 8 x 3 = 24 9 x 3 = 27 Multiples of 3

82. Fill in each column First 8 Multiples of 5First 8 Multiples of 7

83. Click for interactive game practice.

84. 32Select all of the multiples of 6. A B C D 1 42 15 54 E F 35 56

85. 33Select all of the multiples of 9. A B C D 36 9 19 28 E F 76 90

86. 34Select all of the multiples of 4. A B C D 36 25 32 4 E F 22 28

87. 35Select all of the multiples of 8. A B C D 48 78 24 18 E F 62 56

88. Inverse Operations Return to table of contents

89. An inverse operation is the operation that reverses the effect of another operation. Multiplication and division are inverse operations.

90. 7 x 4 = 28 28 ÷ 7 = 4 is the division that undoes the multiplication of 7 x 4 72 ÷ 8 = 9 8 x 9 = 72is the multiplication that undoes the division of 72 ÷ 8

91. Move equations to match each with its inverse. 8 x 3 = 24 24 ÷ 3 = 8 7 x 5 = 35 35 ÷ 7 = 5 6 x 10 = 60 60 ÷ 10 = 6 4 x 6 = 24 24 ÷ 6 = 4

92. 36Division and multiplication are inverse operations. True False

93. 37Which equation shows the inverse operation for the equation 63 ÷ 9 = 7? A B C D 63 - 9 = 54 21 ÷ 3 = 7 7 x 9 = 63 3 x 7 = 63

94. 38Which equation shows the inverse operation for the equation 5 x 4 = 20? A B C D 10 x 2 = 20 20 ÷ 2 = 10 20 ÷ 1 = 20 20 ÷ 4 = 5

95. 39Which equation shows the inverse operation for the equation 6 x 4 = 24? A B C D 24 ÷ 2 = 12 3 x 8 = 24 24 ÷ 6 = 4 24 ÷ 3 = 8

96. 40Which equation shows the inverse operation for the equation 40 ÷ 10 = 4? A B C D 10 x 4 = 40 2 x 20 = 40 40 ÷ 8 = 5 8 x 5 = 40

97. Inverse operations can be used to solve unknowns in an equation. An unknown can be represented using a, ?, or a letter to stand for the missing number. A letter that stands for a missing number in an equation is called a variable.

98. The animal shelter has 18 kittens. The same number of puppies were born to each of 3 mother cats. How many kittens did each mother cat have? 3 x = 18 Use the inverse operation of multiplication to solve. 18 ÷ 3 = 6 Each mother cat had 6 kittens.

99. Use the inverse operation of multiplication to solve. 42 ÷ 7 = 6 $6 per week would need to be saved. A new video game you want is $42. How much money do you need to save per week if you want to buy it in 7 weeks. 7 x ? = 42

100. Use the inverse operation of multiplication to solve. 32 ÷ 4 = 8 There are 8 cards in each group. Mariella has 32 game cards. They are in 4 equal groups on her desk. How many game cards (c) are in each pile? 4 x c = 32

101. 41Use inverse operations to solve for the unknown in the equation. 16 ÷ = 2

102. 42Use inverse operations to solve for the unknown in the equation. 7 x ? = 49

103. 43Use inverse operations to solve for the unknown in the equation. y x 6 = 54

104. 44 Use inverse operations to solve for the unknown in the equation. 36 ÷ ? = 9

105. 45Use inverse operations to solve for the unknown in the equation. 100 ÷ a = 25

106. 46Use inverse operations to solve for the unknown in the equation. x 8 = 48