1. Distributive Property: Advanced Problems
It may be necessary to review the basic distributive property problems in the number property introduction PowerPoint presentation

2. Recall the distributive property of multiplication over addition . . .
symbolically:
a × (b + c) = a × b + a × c
and pictorially (rectangular array area model): b c a
a × b
a × c

3. using your mental math skills . . .
An example: 6 x 13
using your mental math skills . . .
symbolically:
6 × (10 + 3) = 6 × × 3
and pictorially (rectangular array area model): 10 3 6
6 × 10
6 × 3

4. Use the Distributive Property to write as an equivalent expression
Use the Distributive Property to write as an equivalent expression. Then evaluate the expression.
Multiply.
Add.
Answer: 52
Example 1-1a

5. Use the Distributive Property to write as an equivalent expression
Use the Distributive Property to write as an equivalent expression. Then evaluate the expression. ***It doesn’t matter which side of the parenthesis the number is on. The property works the same.
Multiply.
Add.
Answer: 30
Example 1-1b

6. Use the Distributive Property to write each expression as an equivalent expression. Then evaluate the expression. a. b.
Answer:
Answer:
Example 1-1c

7. Method 1 Find the cost for 1 person, then multiply by 4.
Real Life Example: Recreation North Country Rivers of York, Maine, offers one-day white-water rafting trips on the Kennebec River. The trip costs $69 per person, and wet suits are $15 each.
Write two equivalent expressions to find the total cost of one trip for a family of four if each person uses a wet suit.
Method 1 Find the cost for 1 person, then multiply by 4.
cost for 1 person
Example 1-2a

8. Method 2 Find the cost of 4 trips and 4 wet suits. Then add.
cost of 4 wet suits
cost of 4 trips
Example 1-2a

9. Evaluate either expression to find the total cost.
Distributive Property
Multiply.
Add.
Answer: The total cost is $336.
Check You can check your results by evaluating 4($84).
Example 1-2b

10. Movies The cost of a movie ticket is $7 and the cost of a box of popcorn is $2.
a. Write two equivalent expressions to find the total cost for a family of five to go to the movies if each member of the family gets a box of popcorn.
b. Find the total cost.
Answer:
Answer: $45
Example 1-2c

11. Use the Distributive Property to write as an equivalent algebraic expression.
Simplify.
Answer:
Example 1-3a

12. Use the Distributive Property to write as an equivalent algebraic expression.
Simplify.
Answer:
Example 1-3b

13. Use the Distributive Property to write each expression as an equivalent algebraic expression.
Answer:
Answer:
Example 1-3c

14. Distributive Property
Use the Distributive Property to write as an equivalent algebraic expression.
Rewrite as
Distributive Property
Simplify.
Definition of subtraction
Answer:
Example 1-4a

15. Distributive Property
Use the Distributive Property to write as an equivalent algebraic expression.
Rewrite as
Distributive Property
Simplify.
Answer:
Example 1-4b

16. Use the Distributive Property to write each expression as an equivalent algebraic expression.
Answer:
Answer:
Example 1-4c

17. Real-Life Example 2 Mental Math

18. The distributive property is mental math strategy that can be used when multiplying.
43 x 5 =?

19. Break apart the double-digit number.
43 x 5 =? +

20. Then multiply each part by 5.
43 x 5 =? +
x 5 x 5

21. Then multiply each part by 5.
43 x 5 =? +
x 5 x 5

22. Finally, sum your two products
43 x 5 =215 +
x 5 x 5 +
= 215

23. Let’s look at another example.

24. Break apart the double-digit number.
53 x 6 = ?

25. Break apart the double-digit number.
53 x 6 = ? +

26. Multiply each part by 6.
53 x 6 = ? +
x x 6

27. Multiply each part by 6.
53 x 6 = ? +
x x 6

28. Sum the two products.
53 x 6 = 318 +
x x 6
= 318

29. The word “distribute” means “to give out.”

30. Distribute the cubes to the girls.

31. Distribute the cubes to the girls.

32. Distribute the cubes to the girls.

33. Distribute the cubes to the girls.

34. Distribute the cubes to the girls.

35. Distribute the cubes to the girls.

36. In this example, the 5 was distributed.
5 x 38 = 5 x (30 + 8) = (5 x 30) + (5 x 8)

37. In this example, the 7 was distributed.
7 x 46 = 7 x (40 + 6) = (7 x 40) + (7 x 6)

38. Find the area of the rectangle. Area = length x width

39. Find the area of the rectangle. Area = length x width

40. Find the area of the rectangle. Area = length x width
20 ft ft

41. Find the area of the rectangle. Area = length x width
20 ft ft

42. Find the area of the rectangle. Area = length x width
20 ft ft

43. Find the area of the rectangle. Area = length x width
Find the area of each rectangle.
6 ft
6 ft
20 ft ft

44. Find the area of the rectangle. Area = length x width
Find the area of each rectangle.
6 ft
6 ft
6 x 20 = 120 sq ft
20 ft ft

45. Find the area of the rectangle. Area = length x width
Find the area of each rectangle.
6 ft
6 ft
6 x 20 = 120 sq ft
6 x 4 = 24
sq ft
20 ft ft

46. Find the area of the rectangle. Area = length x width
Find the area of each rectangle.
6 ft
6 ft
120 sq ft
24 sq ft
20 ft ft

47. Find the area of the rectangle. Area = length x width
Now put the two rectangles back together.
6 ft
6 ft
120 sq ft
24 sq ft
20 ft ft

48. Find the area of the rectangle. Area = length x width
Now put the two rectangles back together.
6 ft
120 sq ft
24 sq ft
20 ft ft

49. Find the area of the rectangle. Area = length x width
Now put the two rectangles back together.
6 ft
120 sq ft
24 sq ft
20 ft ft

50. Find the area of the rectangle. Area = length x width
Now put the two rectangles back together.
6 ft
120 sq ft
+ 24 sq ft
24 ft

51. Find the area of the rectangle. Area = length x width
Now put the two rectangles back together.
6 ft
144 sq ft
24 ft

52. A swimming pool has a shallow end and a deep end
A swimming pool has a shallow end and a deep end. Find the surface area of the pool.
deepwater
8 yds
shallow water
5 yds
10 yds

53. Break the pool into a deep end and a shallow end.
deepwater
8 yds
8 yds
shallow water
10 yds
5 yds

54. Find the area of the deep end.
deepwater
8 yds
8 yds
shallow water
10 yds
5 yds

55. Find the area of the deep end.
8 x 5 = 40
8 yds
8 yds
shallow water
10 yds
5 yds

56. Find the area of the shallow end.
8 x 5 = 40
8 yds
8 yds
shallow water
10 yds
5 yds

57. Find the area of the shallow end.
8 x 5 = 40
8 yds
8 yds
8 x 10 = 80
10 yds
5 yds

58. Now sum the two areas together.
8 x 5 = 40
8 yds
8 yds
8 x 10 = 80
10 yds
5 yds

59. Now sum the two areas together.40 80 +
8 yds
10 yds
5 yds

60. = 120 square yards 40 80
8 yds
5 yds
10 yds

61. Write an expression that shows how to find the area of the rectangle and uses the distributive property.
9 yds
5 yds
20 yds

62. Find the areas for each individual rectangle.
9 yds
5 yds
20 yds

63. Find the areas for each individual rectangle.
9 yds
(9 x 5)
5 yds
20 yds

64. Find the areas for each individual rectangle.
9 yds
(9 x 5)
(9 x 20)
5 yds
20 yds

65. Sum the two areas.
9 yds +
(9 x 5)
(9 x 20)
5 yds
20 yds

66. (9 x 5) + (9 x 20) = area
9 yds
(9 x 5)
(9 x 20)
5 yds
20 yds

67. Practice Time

68. 1) Which of the following expressions shows the distributive property for 5 x (3 + 7)?
(5 x 3) x (5 x 7)
(5 x 3) + (5 x 7)

69. 1) Which of the following expressions shows the distributive property for 5 x (3 + 7)?
Correct!
(5 x 3) + (5 x 7)

70. 2) Which of the following expressions shows the distributive property for 3 x (9 + 4) ?
(3 x 9) + (3 x 4)
(3 + 9) + (3 + 4)
(3 + 9) x (3 + 4)

71. 2) Which of the following expressions shows the distributive property for 3 x (9 + 4) ?
(3 x 9) + (3 x 4)
Correct!

72. 3) Which of the following expressions is equivalent to: and shows the distributive property.
2 x (2 + 3)
3 x (2 + 3)

73. 3) Which of the following expressions is equivalent to: and uses the distributive property.
2 x (2 + 3)
Correct!

74. 4) Which of the following expressions is equivalent to: (4 x 3) + (4 x 8) ?

75. 4) Which of the following expressions is equivalent to: (4 x 3) + (4 x 8) ?
Correct!
4 x (3 + 8)

76. 5) Which of the following expressions is equivalent to: (5 x 9) + (5 x 3) ?

77. 5) Which of the following expressions is equivalent to: (5 x 9) + (5 x 3) ?
Correct!

78. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
3 yds
9 yds

79. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
3 yds
9 yds

80. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
3 yds
9 yds

81. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
4 yds
3 yds
9 yds

82. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
4 yds
4 x 3
3 yds
9 yds

83. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 yds
4 yds
4 x 3
4 x 9
3 yds
9 yds

84. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 x 3
4 x 9
4 yds
4 yds
3 yds
9 yds

85. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 x 3
4 x 9
4 yds
3 yds
9 yds

86. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 x 3
4 x 9
4 yds
3 yds
9 yds

87. 6) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
4 x 3 +
4 x 9
4 yds
3 yds
9 yds

88. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
4 yds
8 yds

89. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
4 yds
8 yds

90. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
4 yds
8 yds

91. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
6 yds
4 yds
8 yds

92. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
6 yds
6 x 4
4 yds
8 yds

93. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 yds
6 yds
6 x 4
6 x 8
4 yds
8 yds

94. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 x 4
6 x 8
6 yds
6 yds
4 yds
8 yds

95. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 x 4
6 x 8
6 yds
4 yds
8 yds

96. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 x 4
6 x 8
6 yds
4 yds
8 yds

97. 7) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
6 x 4 +
6 x 8
6 yds
4 yds
8 yds

98. 8) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5 yds
2 yds
10 yds

99. 8) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5 yds
5 x 2
5 x 10
2 yds
10 yds

100. 8) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5 x 2 +
5 x 10
5 yds
2 yds
10 yds

101. 9) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
8 yds
3 yds
5 yds

102. 9) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
8 yds
8 x 3
8 x 5
3 yds
5 yds

103. 9) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
8 x
8 x 5
8 yds
3 yds
5 yds

104. 10) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5 yds
x yds
10 yds

105. 10) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5 yds 5x
5 ∙ 10
x yds
10 yds

106. 10) Write an expression that shows how to find the area of the rectangle and uses the distributive property.
5x +
5 ∙ 10
5 yds
x yds
10 yds

107. 11) Which expression is equivalent to 3(x + 7)?

108. 11) Which expression is equivalent to 3(x + 7)?
Correct!
3x + 21

109. 12) Which expression is equivalent to 4(x + 5)?

110. 12) Which expression is equivalent to 4(x + 5)?
Correct!

111. 13) Which expression is equivalent to 8(x + 2)?

112. 13) Which expression is equivalent to 8(x + 2)?
Correct!

113. 14) Which expression is equivalent to 2(x + 3)?

114. 14) Which expression is equivalent to 2(x + 3)?
Correct!

115. Click below to see video

116. Click to Test Your Skills