- The document is from a presentation on fractions given on April 27, 2013 by Joan Cotter. - It discusses why fractions are important to learn, such as for sharing pizza, cooking, reading rulers, and preparing for algebra. - It includes examples of using fractions in comics and charts showing fraction relationships. Games are presented to help students understand unit fractions and that combinations of fractions can make a whole.

1. Fractions
MassHOPE-TEACH
Worcester, MA
Saturday, April 27, 2013
2:45pm - 3:45pm
Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com

2. Why Learn Fractions
• Sharing pizza

3. Why Learn Fractions
• Sharing pizza
• Cooking and baking

4. Why Learn Fractions
• Sharing pizza
• Cooking and baking
• Reading rulers

5. Why Learn Fractions
• Sharing pizza
• Cooking and baking
• Reading rulers
• Easing into decimals
• Learning algebra

6. Fractions in the Comics

7. Fractions in the Comics

8. Fraction Chart
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

9. Fraction Chart

10. Fraction Stairs
1

11. Fraction Stairs
1
1
2

12. Fraction Stairs
1
1
2
1
3

13. Fraction Stairs
1
1
2
1
3
1
4

14. Fraction Stairs
1
1
2
1
3
1
4
1
5

15. Fraction Stairs
1
1
2
1
3
1
4
1
5
1
6

16. Fraction Stairs
1
1
2
1
3
1
4
1
5
1
7
1
6

17. Fraction Stairs
1
1
2
1
3
1
4
1
5
1
7
1
8
1
6

18. Fraction Stairs
1
1
2
1
3
1
4
1
5
1
7
1
8
1
6
1
9

19. Fraction Stairs
1
1
2
1
3
1
4
1
5
1
7
1
8
1
10
1
6
1
9

20. Fraction Stairs

21. Fraction Stairs

22. Fraction Stairs
A hyperbola.

23. Fraction Names
• In English, except for half, we use
ordinal numbers to name fractions.

24. Sometimes called quarter.
Fourths

25. Sometimes called quarter.
• A quarter of a hour (15 min.)
Fourths

26. Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
Fourths

27. Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
• A quarter of a gallon (quart)
Fourths

28. Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
• A quarter of a gallon (quart)
• A Quarter Pounder (4 oz.)
Fourths

29. Writing Fractions
1 one

30. Writing Fractions
1 one
divided by

31. Writing Fractions
1
3
one
divided by
three

32. Writing Fractions
1
3
one
divided by
three
1
1
3
1
3
1
3

33. Writing Fractions
1
3
one
divided by
three
Avoid saying “over” as in 1 over 3.
1
1
3
1
3
1
3

34. Unit Fraction War
Objective:
To help the children realize a unit
fraction decreases as the denominator
increases.

35. Unit Fraction War
Object of the game:
To collect all, or most, of the cards
with the greater unit fraction.
Objective:
To help the children realize a unit
fraction decreases as the denominator
increases.

36. Unit Fraction War

37. 1
5
1
4
Unit Fraction War

38. 1
5
1
4
Unit Fraction War

39. Unit Fraction War

40. Unit Fraction War
11
8

41. Unit Fraction War
11
8

42. Unit Fraction War

43. Unit Fraction War
1
6
1
6

44. Unit Fraction War
1
6
1
6

45. Unit Fraction War
1
6
1
6
1
3
1
4

46. Unit Fraction War

47. Fraction Chart
How many fourths in a
whole?
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

48. Fraction Chart
How many fourths in a
whole?
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
4
1
4
1
2
1
1
3

49. 1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths in a
whole?
1
4
1
4
1
2
1
1
3

50. Fraction Chart
How many fourths in a
whole?
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
4
1
4
1
2
1
1
3
1
4
1
2

51. Fraction Chart
How many fourths in a
whole?
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
7
1
8
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
4
1
4
1
2
1
1
3
1
2
1
3
1
4
1
4

52. Fraction Chart
How many sixths in a
whole?
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

53. Fraction Chart
How many eighths in a
whole?
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

54. Concentrating on One Game
Objective:
To help the children realize that 5 fifths, 8
eighths, and so forth, make a whole.

55. Concentrating on One Game
Object of the game:
To find the pairs that make a whole.
Objective:
To help the children realize that 5 fifths, 8
eighths, and so forth, make a whole.

56. Concentrating on One

57. Concentrating on One
5
3

58. Concentrating on One
5
3

59. Concentrating on One
5
3

60. Concentrating on One
5
3
2
5

61. Concentrating on One

62. Concentrating on One
3
8

63. Concentrating on One
3
8

64. Concentrating on One
3
8

65. Concentrating on One
3
8
7
8

66. Concentrating on One

67. Concentrating on One
5
8

68. Concentrating on One
5
8

69. Concentrating on One
5
8

70. Concentrating on One
3
8
5
8

71. Concentrating on One

72. Concentrating on One

73. Fraction Chart
Which is more, 3/4 or
4/5?
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

74. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
Which is more, 3/4 or
4/5?

75. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
Which is more, 3/4 or
4/5?

76. Fraction Chart
Which is more, 7/8 or
8/9?
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

77. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
Which is more, 7/8 or
8/9?

78. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
An interesting pattern.

79. Partial Chart
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

80. Partial Chart
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

81. Partial Chart

82. Partial Chart
1 2 3 4 5 6

83. Fraction War

84. Fraction War
Objective:
To practice comparing ones, halves,
fourths, and eighths in preparation for
reading a ruler.

85. Fraction War
Object of the game:
To capture all the cards.
Objective:
To practice comparing ones, halves,
fourths, and eighths in preparation for
reading a ruler.

86. Fraction War
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

87. Fraction War
1
4
1
8
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

88. Fraction War
1
4
1
8
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

89. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Fraction War
1
4
1
8

90. Fraction War
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

91. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Fraction War
3
4
5
8

92. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Fraction War
3
4
5
8

93. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Fraction War
3
4
5
8

94. Fraction War
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

95. Fraction War
3
4
3
4
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

96. Fraction War
3
4
3
4
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

97. Fraction War
3
4
3
4
3
8
1
4
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

98. Fraction War
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

99. Fraction War
1
1
4
1
4
1
4
1
4
1
2
1
2
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8

100. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half?

101. 1
1
2
1
2
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half?

102. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half?

103. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half? Eighths?

104. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half? Eighths?

105. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half? Eighths?
Sevenths?

106. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
How many fourths equal a half? Eighths?
Sevenths?

107. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/4 plus 1/8?

108. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/4 plus 1/8?

109. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/4 plus 1/8?

110. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/4 plus 1/8? [3/8]

111. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is
1/4.

112. 1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is
1/4.

113. Faulty Fractions

114. Faulty Fractions
“Fish Tank” model

115. Faulty Fractions
2
“Fish Tank” model

116. Faulty Fractions
2
5
“Fish Tank” model

117. Faulty Fractions
= part
whole
CRA model

118. Faulty Fractions
= part
whole
“Goal: To develop the spatial organization, visually
and kinesthetically, to read and write fractions
correctly.
CRA model

119. Faulty Fractions
= part
whole
“Goal: To develop the spatial organization, visually
and kinesthetically, to read and write fractions
correctly.
“Materials: Red squares and larger black squares are
displayed to help with sequencing and number
placement.”
CRA model

120. Faulty Fractions
This is fourths.
“Words” model

121. Faulty Fractions
This is fourths. This is thirds.
“Words” model

122. Faulty Fractions
1
3
1
3
1
3
1
4
1
4
1
4
1
4
“Rounded corners”

123. Faulty Fractions
The middle fractions are greater
than the fractions at the ends!
1
3
1
3
1
3
1
4
1
4
1
4
1
4
“Rounded corners”

124. Faulty Fractions
1
2
1
1
2
1
3
1
3
1
3
6 6 6 6 6 6
1
7
1
7
1
7
1
7
1
7
1
7
1
7
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
4
1
4
1
4
1
9
1
9
1
9
1
9
1
9
1
9
1
9
1
9
1
5
1
5
1
5
1
5
1
5
1 1 1 1 1 1
1
4
“Color” model

125. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
3
1
3
1
3
1
5
1
5
1
5
1
5
1
5
1
6
1
6
1
6
1
6
1
6
1
6
Faulty Fractions
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
1
12
Missing 7ths & 9ths

126. 1
12
Faulty Fractions
Missing 7ths & 9ths
1
1
2
1
3
1
4
1
5
1
8
1
10
1
6

127. Faulty Fractions
Are we comparing angles, arcs, or area?
Circles

128. Faulty Fractions
6
1
6
1
6
1
6
1
6
1
6
1
5
1
4
1
2
1
3
1
5
1
5
1
5
1
5
1
4
1
4
1
4
1
3
1
3
1
2
1
Try to compare 4/5 and 5/6 with this model.
Circles

129. Faulty Fractions
Experts in visual literacy say that
comparing quantities in pie charts is
difficult because most people think
linearly. It is easier to compare along a
straight line than compare pie slices.
askoxford.com
Circles

130. Faulty Fractions
Experts in visual literacy say that
comparing quantities in pie charts is
difficult because most people think
linearly. It is easier to compare along a
straight line than compare pie slices.
askoxford.com
Specialists also suggest refraining from
using more than one pie chart for
comparison.
www.statcan.ca
Circles

131. Definition of a Fraction
What is the definition of a fraction?

132. Definition of a Fraction
What is the definition of a fraction?
A part of a set or part of a whole, a small part.

133. Definition of a Fraction
What is the definition of a fraction?
A part of a set or part of a whole, a small part.
This is the everyday meaning of fraction.

134. Definition of a Fraction
3
2What about ?
What is the definition of a fraction?
A part of a set or part of a whole, a small part.
This is the everyday meaning of fraction.

135. Definition of a Fraction
An expression that indicates
the
quotient of two quantities.
American Heritage Dictionary:

136. Definition of a Fraction
An expression that indicates
the
quotient of two quantities.
This is the mathematical meaning of fraction.
American Heritage Dictionary:

137. Definition of a Fraction
This is the mathematical meaning of fraction.
3
2
An expression that indicates
the
quotient of two quantities.
American Heritage Dictionary:

138. Fractions > 1
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10

139. Fractions > 1
1
1
2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
8

140. Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number
and as an improper fraction.

141. Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number
and as an improper fraction.

142. Mixed to Improper Fractions
Each row of connected rectangles represents 1.
2 =3
4
11
4
Write each quantity as a mixed number
and as an improper fraction.

143. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
Each row of connected rectangles represents 1.
two 4s

144. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
24
two 4s
Each row of connected rectangles represents 1.

145. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
24
Each row of connected rectangles represents 1.
two 4s + 3

146. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
44
3
Each row of connected rectangles represents 1.
two 4s + 3
2

147. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
44
3
Each row of connected rectangles represents 1.
two 4s + 3 = 11
2

148. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
44
3 11
Each row of connected rectangles represents 1.
two 4s + 3 = 11
2

149. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
4 =2
3
14
34
113
Each row of connected rectangles represents 1.
two 4s + 3 = 11
2

150. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
4 =2
3
14
34
113
Each row of connected rectangles represents 1.
two 4s + 3 = 11
four 3s + 2 = 14
2

151. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
4 =2
3
14
3
4 =3
5
23
5
Each row of connected rectangles represents 1.
4
113
two 4s + 3 = 11
four 3s + 2 = 14
2

152. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
4 =2
3
14
3
4 =3
5
23
5
Each row of connected rectangles represents 1.
4
113
four 3s + 2 = 14
two 4s + 3 = 11
2

153. Write each quantity as a mixed number
and as an improper fraction.
Mixed to Improper Fractions
2 =3
4
11
4
4 =2
3
14
3
4 =3
5
23
5
Each row of connected rectangles represents 1.
24
113
four 3s + 2 = 14
two 4s + 3 = 11
four 5s + 3 = 23

154. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.

155. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.
= 211
5
1
5

156. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.
= 211
5
1
5

157. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.
= 211
5
1
5

158. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.
= 211
5
1
5
= 15
3
2
3

159. Improper to Mixed Fractions
Circle the wholes and write each quantity as
an improper fraction and as a mixed number.
= 211
5
1
5
= 15
3
2
3

160. Fraction of Geometric Figures
1
2
Shade

161. Fraction of Geometric Figures
1
2
Shade

162. Fraction of Geometric Figures
1
2
2
3
Shade Shade

163. Fraction of Geometric Figures
1
2
2
3
Shade Shade

164. Fraction of Geometric Figures
1
2
2
3
1
4
Shade Shade Shade

165. Fraction of Geometric Figures
1
2
2
3
1
4
Shade Shade Shade

166. Making the Whole
Draw the whole.
1
3

167. Making the Whole
Draw the whole.
1
3
1
3
1
3
1
3

168. Making the Whole
Draw the whole.
1
3
1
3
1
3
1
3
2
3

169. Making the Whole
Draw the whole.
1
3
1
3
1
3
1
3
2
3
2
3
1
3

170. 1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/2 of 1/2?
1
2

171. 1
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/2 of 1/2?
1
2
1
4

172. 1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
Fraction Chart
What is 1/3 of 1/2?
1
2

173. Fraction Chart
What is 1/3 of 1/2?
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
1
3
1
3
1
4
1
5
1
7
1
8
1
9
1
4
1
5
1
6
1
7
1
8
1
4
1
5
1
6
1
7
1
8
1
9
1
5
1
6
1
6
1
7
1
7
1
7
1
8
1
8
1
8
1
8
1
9
1
9
1
9
1
9
1
9
1
9
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
2
1
6

174. Simplifying Fractions
1
2

175. Simplifying Fractions
3
6
= 1
2

176. Simplifying Fractions
4
8
= 1
2

177. Simplifying Fractions

178. Simplifying Fractions
9
12

179. Simplifying Fractions
9
12 3

180. Simplifying Fractions
9
12 3

181. Simplifying Fractions
9
12
= 3
43

182. Simplifying Fractions

183. Simplifying Fractions

184. Simplifying Fractions

185. Simplifying Fractions

186. Simplifying Fractions
21
28

187. Simplifying Fractions
21
28

188. Simplifying Fractions
21
28

189. Simplifying Fractions
45
72

190. Simplifying Fractions
45
72

191. Simplifying Fractions
45
72

192. Simplifying Fractions
12
16

193. Simplifying Fractions
12
16

194. Simplifying Fractions
12
16

195. Simplifying Fractions
12
16

196. Simplifying Fractions
12
16

197. Simplifying Fractions
12
16

198. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Twos
2 4 6 8 10
12 14 16 18 20

199. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Twos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second
row.

200. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Fours
4 8 12 16 20
24 28 32 36 40
The ones repeat in the second
row.

201. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80

202. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80

203. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80

204. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples

205. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples

206. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples

207. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples

208. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples

209. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
6 x 4
6 x 4 is the fourth number

210. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80 8 x 7
8 x 7 is the seventh number

211. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Nines
9 18 27 36 45
90 81 72 63 54
The second row is written in reverse
order.
Also the digits in each number add to
9.

212. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

213. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

214. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

215. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

216. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

217. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

218. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

219. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

220. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

221. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

222. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Observe the ones.

223. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
The tens are the same in each row.

224. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the digits in the columns.

225. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the digits in the columns.

226. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the digits in the columns.

227. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the “opposites.”

228. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the “opposites.”

229. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the “opposites.”

230. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Threes
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns:
Add the “opposites.”

231. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.

232. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.

233. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.

234. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.

235. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.

236. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.

237. © Joan A. Cotter, Ph.D., 2013
Multiples Patterns
Sevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.

238. Subtracting Fractions
4684
–2372
Preliminary understanding

239. Subtracting Fractions
4684
–2372
2000
Preliminary understanding

240. Subtracting Fractions
4684
–2372
2000
300
Preliminary understanding

241. Subtracting Fractions
4684
–2372
2000
300
10
Preliminary understanding

242. Subtracting Fractions
4684
–2372
2000
300
10
2
Preliminary understanding

243. Subtracting Fractions
4684
–2372
2000
300
10
2
2312
Preliminary understanding

244. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
Preliminary understanding

245. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
2000
Preliminary understanding

246. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
2000
–200
Preliminary understanding

247. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
2000
–200
10
Preliminary understanding

248. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
2000
–200
10
–5
Preliminary understanding

249. Subtracting Fractions
4684
–2372
4684
–2879
2000
300
10
2
2312
2000
–200
10
–5
1805
Preliminary understanding

250. Subtracting Fractions
3
5
4
–

251. Subtracting Fractions
3
5
4
–
32
5

252. Subtracting Fractions
3
5
4
–
32
5
5
7
5
– 2
1
7

253. Subtracting Fractions
3
5
4
–
32
5
3
5
7
5
– 2
1
7

254. Subtracting Fractions
3
5
4
–
32
5
3
– 4
7
5
7
5
– 2
1
7

255. Subtracting Fractions
3
5
4
–
32
5
3
– 4
7
23
7
5
7
5
– 2
1
7

256. Multiplying Fractions

257. Multiplying Fractions
• Multiplication is more than repeated
addition.

258. Multiplying Fractions
4 x 4 = 4 + 4 + 4 + 4
• Multiplication is more than repeated
addition.

259. Multiplying Fractions
4 x 4 = 4 + 4 + 4 + 4
• Repeated addition doesn’t work well with
fractions.
• Multiplication is more than repeated
addition.

260. Multiplying Fractions
• Repeated addition doesn’t work well with
fractions.
1
2
x = + ?
1
2
1
2
4 x 4 = 4 + 4 + 4 + 4
• Multiplication is more than repeated
addition.

261. Multiplying Fractions
Area is a better
model.
4 x 4 =

262. Multiplying Fractions
1
2
x =
1
2
The square represents 1.

263. Multiplying Fractions
1
2
x =
1
2

264. Multiplying Fractions
1
2
x =
1
2
1
4
The solution is the double-crosshatched
area.

265. Multiplying Fractions
2
3
x =
3
4

266. Multiplying Fractions
2
3
x =
3
4

267. Multiplying Fractions
2
3
x =
3
4

268. Multiplying Fractions
2
3
x =
3
4
6
12

269. Multiplying Fractions
x =
3
4
1
2
=
2
3
6
12

270. Multiplying Fractions
2
3
x =
3
4
The total number of rectangles is 3 x 4.

271. Multiplying Fractions
2
3
3
4
The number of double-crosshatched rectangles is 2
The total number of rectangles is 3 x 4.
x =

272. Multiplying Fractions
2
3
3
4
This is why we multiply fractions by
multiplying numerators and denominators.
x =

273. Dividing Fractions

274. Dividing Fractions
÷ =
1
2
1

275. Dividing Fractions
÷ =
1
2
1
1 ÷ 1/2 means how many 1/2s in 1.

276. Dividing Fractions
÷ =
1
2
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/2 means how many 1/2s in 1.

277. Dividing Fractions
÷ =
1
2
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/2 means how many 1/2s in 1.

278. Dividing Fractions
÷ =
1
2
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/2 means how many 1/2s in 1.

279. Dividing Fractions
÷ =
1
2
1 2
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/2 means how many 1/2s in 1.

280. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/3 means how many 1/3s in 1.

281. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/3 means how many 1/3s in 1.

282. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/3 means how many 1/3s in 1.

283. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/3 means how many 1/3s in 1.

284. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 1/3 means how many 1/3s in 1.

285. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3

286. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1 ÷ 2/3 means how many 2/3s in 1.

287. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
2
3
2
3
1 ÷ 2/3 means how many 2/3s in 1.

288. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
2
3
2
3
1 ÷ 2/3 means how many 2/3s in 1.

289. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
2
3
2
3
3
2
2
3
1 ÷ 2/3 means how many 2/3s in 1.

290. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
2
3
2
3
3
2
2
3
1 ÷ 2/3 also must be half of 1 ÷ 1/3.

291. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =1 3
÷ =
2
3
1 3
2
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
2
3

292. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =1 3
÷ =
2
3
1 3
2
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
2
3
1 ÷ 3 is simply the definition of a fraction.

293. 1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
1
3
÷ =1 3
÷ =
2
3
1 3
2
2
3
1 ÷ 3 is simply the definition of a fraction.

294. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1
÷ =1 3
÷ =1 4
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
3
2
2
3
1
3

295. Dividing Fractions
÷ =
1
2
1
÷ =
1
3
1 1
4
÷ =1 3
÷ =1 4
÷ =
2
3
1
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
3
2
2
3
1
3

296. Dividing Fractions
1
3
1
4
÷ =1 3
÷ =1 4
÷ =1 4
3
1
2
1
1
3
1
÷ =
2
3
1
÷ =
÷ =
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
3
2
2
3

297. Dividing Fractions
÷ =1 3
÷ =1 4
1
1
2
1
2
1
3
1
3
1
3
1
4
1
4
1
4
1
4
÷ =1 4
3
1
2
1
1
3
1
÷ =
2
3
1
÷ =
÷ =
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1
3
1
3
1
4
3
2
2
3

298. Dividing Fractions
÷ =1 3
÷ =1 4
1
1
2
1
2
1
3
1
3
1
3
1
4
1
4
1
4
1
4
÷ =1 4
3
1
2
1
1
3
1
÷ =
2
3
1
÷ =
÷ =
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1
3
1
3
1
4
3
2
2
3
Only 3/4 of the 4/3 fits into the 1.

299. Dividing Fractions
÷ =1 3
÷ =1 4
1
1
2
1
2
1
3
1
3
1
3
1
4
1
4
1
4
1
4
÷ =1 4
3
1
2
1
1
3
1
÷ =
2
3
1
÷ =
÷ =
1
1
2
1
2
1
4
1
4
1
4
1
4
1
3
1
3
1
3
1
3
1
3
1
4
3
2
2
3
3
4
Only 3/4 of the 4/3 fits into the 1.

300. Dividing Fractions
÷ =1 3
÷ =1 4
÷ =1 4
3
1
2
1
1
3
1
÷ =
2
3
1
÷ =
÷ =
1
3
1
4
3
2
2
3
3
4

301. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
÷ =1 4
3
4
÷ =1 4
3
÷ =
2
3
1 3
2

302. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
÷ =1 4
3
4
÷ =1 4
3
÷ =
2
3
1 3
2
The colored pairs are reciprocals of each
other.
A reciprocal may be called a multiplicative
inverse.

303. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
÷ =1 4
3
4
÷ =1 4
3
÷ =
2
3
1 3
2
The colored pairs are reciprocals of each
other.
A reciprocal may be called a multiplicative
inverse. When multiplied together, they
equal 1.

304. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
÷ =1 4
3
4
÷ =1 4
3
÷ =
2
3
1 3
2
The colored pairs are reciprocals of each
other.
A reciprocal may be called a multiplicative
inverse. When multiplied together, they
equal 1.
In the equation 6 ÷ 2 = 3, 6 = 2 x 3.

305. Dividing Fractions
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
1
÷ =1 4
1
3
4
÷ =1 4
3
÷ =
2
3
1 3
2

306. Dividing Fractions
Sometimes textbooks put a 1 under a
whole number to make it look like a
fraction, but it is not necessary.
÷ =
1
2
1 2
÷ =
1
3
1 3
1
3
1
4
÷ =1 3
1
÷ =1 4
1
3
4
÷ =1 4
3
÷ =
2
3
1 3
2

307. Dividing Fractions
÷ = __
2
3
5To find

308. Dividing Fractions
÷ = __
2
3
5To find
First think about finding 1 ÷ 2/3.

309. Dividing Fractions
÷ = __
2
3
5
÷ =
2
3
1First find
To find

310. Dividing Fractions
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find

311. Dividing Fractions
÷ =
2
3
5
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then

312. Dividing Fractions
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then ÷ =
2
3
5 5 (1 )x ÷
2
3

313. Dividing Fractions
= x =
3
25
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then ÷ =
2
3
5 5 (1 )x ÷
2
3

314. Dividing Fractions
= x =
3
2
15
25
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then ÷ =
2
3
5 5 (1 )x ÷
2
3

315. Dividing Fractions
= x = =
3
2
1
25 7
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then ÷ =
2
3
5 5 (1 )x ÷
2
3
15
2

316. Dividing Fractions
= x = =
3
2
1
25 7
÷ = __
2
3
5
÷ =
2
3
1 3
2
First find
To find
Then ÷ =
2
3
5 5 (1 )x ÷
2
3
15
2
Does the answer make sense?
About how many 2/3s are in 5?

317. 2
3
Dividing Fractions
÷ = __Find
3
4

318. 2
3
Dividing Fractions
÷ = __Find
3
4
(Is the answer more or less than 1?)

319. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
3
1
3
1
3
2
3
Dividing Fractions
÷ = __Find
3
4
(Is the answer more or less than 1?)

320. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
3
1
3
1
3
2
3
Dividing Fractions
÷ = __Find
3
4
(Is the answer more or less than 1?)

321. 1
1
4
1
4
1
4
1
4
1
2
1
2
1
3
1
3
1
3
2
3
Dividing Fractions
÷ = __Find
3
4
(The answer must be less than 1.)

322. Dividing Fractions
To find
2
3
÷ = __
3
4
÷ =
3
4
1First find

323. Dividing Fractions
To find
2
3
÷ = __
3
4
÷ =
3
4
1 4
3
First find

324. Dividing Fractions
÷ =
3
4
1 4
3
First find
To find
Then
÷ = __
3
4
2
3
÷ =
3
4
2
3

325. Dividing Fractions
÷ =
3
4
1 4
3
First find
To find
Then
÷ = __
3
4
2
3
3
4
2
3
3
4
2
3
÷ = x (1 ÷ )

326. Dividing Fractions
÷ =
3
4
1 4
3
First find
To find ÷ = __
3
4
2
3
Then
3
4
2
3
3
4
2
3
÷ = x (1 ÷ )
= x =
4
3
2
3

327. = x =
Dividing Fractions
÷ =
3
4
1 4
3
First find
To find ÷ = __
3
4
2
3
Then
3
4
2
3
3
4
2
3
÷ = x (1 ÷ )
8
9
4
3
2
3

328. = x =
Dividing Fractions
÷ =
3
4
1 4
3
First find
To find ÷ = __
3
4
2
3
Then
3
4
2
3
3
4
2
3
÷ = x (1 ÷ )
8
9
4
3
2
3
The answer should be < 1 and it is.

329. = x =
Dividing Fractions
÷ =
3
4
1 4
3
First find
To find ÷ = __
3
4
2
3
Then
3
4
2
3
3
4
2
3
÷ = x (1 ÷ )
8
9
4
3
2
3
The extra step of dividing by 1 can be omitted.

330. ÷ = x (1 ÷ )
= x =
Dividing Fractions
÷ =
3
4
1 4
3
First find
To find ÷ = __
3
4
2
3
Then
3
4
2
3
3
4
2
3
8
9
4
3
2
3
The extra step of dividing by 1 can be omitted.

331. Dividing Fractions
It’s ours to reason why
We invert and multiply.

332. Fraction Meanings

333. Fraction Meanings
• One or more equal parts of a whole.

334. Fraction Meanings
• One or more equal parts of a whole.
• One or more equal parts of a collection.

335. Fraction Meanings
• One or more equal parts of a whole.
• One or more equal parts of a collection.
• Division of two whole numbers.

336. Fraction Meanings
• One or more equal parts of a whole.
• One or more equal parts of a collection.
• Location on a number line.
• Division of two whole numbers.

337. Fraction Meanings
• One or more equal parts of a whole.
• Ratio of two numbers.
• One or more equal parts of a collection.
• Location on a number line.
• Division of two whole numbers.

338. Fractions
MassHOPE-TEACH
Worcester, MA
Saturday, April 27, 2013
2:45pm - 3:45pm
Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com