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# IMF: Visualizing and Montessori Math PART 2

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### IMF: Visualizing and Montessori Math PART 2

1. 1. How Visualization Enhances Montessori Mathematics PART 2 by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com Montessori Foundation Conference Friday, Nov 2, 2012 Sarasota, Florida 8 16 24 32 40 3 2 5 5PowerPoint Presentation & Handout RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
2. 2. Multiplication Strategies Basic facts © Joan A. Cotter, Ph.D., 2012
3. 3. Multiplication Strategies Basic facts6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
4. 4. Multiplication Strategies Basic facts6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
5. 5. Multiplication Strategies Basic facts6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
6. 6. Multiplication Strategies Basic facts6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
7. 7. Multiplication Strategies Basic facts6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
8. 8. Multiplication Strategies Basic facts9× 3= © Joan A. Cotter, Ph.D., 2012
9. 9. Multiplication Strategies Basic facts9× 3= © Joan A. Cotter, Ph.D., 2012
10. 10. Multiplication Strategies Basic facts9× 3=30 © Joan A. Cotter, Ph.D., 2012
11. 11. Multiplication Strategies Basic facts9× 3=30 – 3 = 27 © Joan A. Cotter, Ph.D., 2012
12. 12. Multiplication Strategies Basic facts4× 8= © Joan A. Cotter, Ph.D., 2012
13. 13. Multiplication Strategies Basic facts4× 8= © Joan A. Cotter, Ph.D., 2012
14. 14. Multiplication Strategies Basic facts4× 8= © Joan A. Cotter, Ph.D., 2012
15. 15. Multiplication Strategies Basic facts4× 8=20 + 12 = 32 © Joan A. Cotter, Ph.D., 2012
16. 16. Multiplication Strategies Basic facts7× 7= © Joan A. Cotter, Ph.D., 2012
17. 17. Multiplication Strategies Basic facts7× 7= © Joan A. Cotter, Ph.D., 2012
18. 18. Multiplication Strategies Basic facts7× 7=25 © Joan A. Cotter, Ph.D., 2012
19. 19. Multiplication Strategies Basic facts7× 7=25 + 10 + 10 © Joan A. Cotter, Ph.D., 2012
20. 20. Multiplication Strategies Basic facts7× 7=25 + 10 + 10+ 4 = 49 © Joan A. Cotter, Ph.D., 2012
21. 21. Multiplication on the AL Abacus Commutative property 5× 6=6× 5 © Joan A. Cotter, Ph.D., 2012
22. 22. Multiplication on the AL Abacus Commutative property 5× 6=6× 5 © Joan A. Cotter, Ph.D., 2012
23. 23. Multiplication on the AL Abacus Commutative property 5× 6=6× 5 © Joan A. Cotter, Ph.D., 2012
24. 24. Multiplication on the AL Abacus Commutative property 5× 6=6× 5 © Joan A. Cotter, Ph.D., 2012
25. 25. Multiplication on the AL Abacus For facts > 5 × 57×8= This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
26. 26. Multiplication on the AL Abacus For facts > 5 × 57×8= This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
27. 27. Multiplication on the AL Abacus For facts > 5 × 57×8= Tens: This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
28. 28. Multiplication on the AL Abacus For facts > 5 × 57×8= Tens: This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
29. 29. Multiplication on the AL Abacus For facts > 5 × 57×8= Tens: 20 + 30 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
30. 30. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + Tens: 20 + 30 50 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
31. 31. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + Tens: 20 Ones: + 30 50 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
32. 32. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + Tens: 20 Ones: + 30 50 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
33. 33. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + Tens: 20 Ones: 3 + 30 ×2 50 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
34. 34. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + 6 Tens: 20 Ones: 3 + 30 ×2 50 6 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
35. 35. Multiplication on the AL Abacus For facts > 5 × 57×8=50 + 6 = 56 Tens: 20 Ones: 3 + 30 ×2 50 6 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5. © Joan A. Cotter, Ph.D., 2012
36. 36. Multiplication on the AL Abacus For facts > 5 × 59×7= © Joan A. Cotter, Ph.D., 2012
37. 37. Multiplication on the AL Abacus For facts > 5 × 59×7= © Joan A. Cotter, Ph.D., 2012
38. 38. Multiplication on the AL Abacus For facts > 5 × 59×7= Tens: © Joan A. Cotter, Ph.D., 2012
39. 39. Multiplication on the AL Abacus For facts > 5 × 59×7= Tens: © Joan A. Cotter, Ph.D., 2012
40. 40. Multiplication on the AL Abacus For facts > 5 × 59×7= Tens: 40 + 20 © Joan A. Cotter, Ph.D., 2012
41. 41. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + Tens: 40 + 20 60 © Joan A. Cotter, Ph.D., 2012
42. 42. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + Tens: 40 Ones: + 20 60 © Joan A. Cotter, Ph.D., 2012
43. 43. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + Tens: 40 Ones: + 20 60 © Joan A. Cotter, Ph.D., 2012
44. 44. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + Tens: 40 Ones: 1 + 20 ×3 60 © Joan A. Cotter, Ph.D., 2012
45. 45. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + 3 Tens: 40 Ones: 1 + 20 ×3 60 3 © Joan A. Cotter, Ph.D., 2012
46. 46. Multiplication on the AL Abacus For facts > 5 × 59×7=60 + 3 = 63 Tens: 40 Ones: 1 + 20 ×3 60 3 © Joan A. Cotter, Ph.D., 2012
47. 47. The Multiplication Board 1 2 3 4 5 6 7 8 9 106×4 6 © Joan A. Cotter, Ph.D., 2012
48. 48. The Multiplication Board 1 2 3 4 5 6 7 8 9 106×4 6 © Joan A. Cotter, Ph.D., 2012
49. 49. The Multiplication Board 1 2 3 4 5 6 7 8 9 107×7 7 © Joan A. Cotter, Ph.D., 2012
50. 50. The Multiplication Board 1 2 3 4 5 6 7 8 9 107×7 7 © Joan A. Cotter, Ph.D., 2012
51. 51. The Multiplication Board7×7 © Joan A. Cotter, Ph.D., 2012
52. 52. Multiples Patterns Twos2 4 6 8 1012 14 16 18 20 © Joan A. Cotter, Ph.D., 2012
53. 53. Multiples Patterns Twos2 4 6 8 1012 14 16 18 20 The ones repeat in the second row. © Joan A. Cotter, Ph.D., 2012
54. 54. Multiples Patterns Fours4 8 12 16 2024 28 32 36 40 The ones repeat in the second row. © Joan A. Cotter, Ph.D., 2012
55. 55. Multiples Patterns Sixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 © Joan A. Cotter, Ph.D., 2012
56. 56. Multiples Patterns Sixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 © Joan A. Cotter, Ph.D., 2012
57. 57. Multiples Patterns Sixes and Eights6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 © Joan A. Cotter, Ph.D., 2012
58. 58. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80The ones in the 8s show the multiples of 2. © Joan A. Cotter, Ph.D., 2012
59. 59. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80The ones in the 8s show the multiples of 2. © Joan A. Cotter, Ph.D., 2012
60. 60. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80The ones in the 8s show the multiples of 2. © Joan A. Cotter, Ph.D., 2012
61. 61. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80The ones in the 8s show the multiples of 2. © Joan A. Cotter, Ph.D., 2012
62. 62. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80The ones in the 8s show the multiples of 2. © Joan A. Cotter, Ph.D., 2012
63. 63. Multiples Patterns Sixes and Eights6 12 18 24 30 6× 436 42 48 54 608 16 24 32 4048 56 64 72 806 × 4 is the fourth number (multiple). © Joan A. Cotter, Ph.D., 2012
64. 64. Multiples Patterns Sixes and Eights 6 12 18 24 3036 42 48 54 60 8 16 24 32 4048 56 64 72 80 8× 78 × 7 is the seventh number (multiple). © Joan A. Cotter, Ph.D., 2012
65. 65. Multiples Patterns Nines 9 18 27 36 4590 81 72 63 54The second row is written in reverse order.Also the digits in each number add to 9. © Joan A. Cotter, Ph.D., 2012
66. 66. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
67. 67. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
68. 68. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
69. 69. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
70. 70. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
71. 71. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
72. 72. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
73. 73. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
74. 74. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
75. 75. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
76. 76. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Observe the ones. © Joan A. Cotter, Ph.D., 2012
77. 77. 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. © Joan A. Cotter, Ph.D., 2012
78. 78. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns:Add the digits in the columns. © Joan A. Cotter, Ph.D., 2012
79. 79. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns:Add the digits in the columns. © Joan A. Cotter, Ph.D., 2012
80. 80. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns:Add the digits in the columns. © Joan A. Cotter, Ph.D., 2012
81. 81. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Add the “opposites.” © Joan A. Cotter, Ph.D., 2012
82. 82. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Add the “opposites.” © Joan A. Cotter, Ph.D., 2012
83. 83. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Add the “opposites.” © Joan A. Cotter, Ph.D., 2012
84. 84. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30The 3s have several patterns: Add the “opposites.” © Joan A. Cotter, Ph.D., 2012
85. 85. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70The 7s have the 1, 2, 3… pattern. © Joan A. Cotter, Ph.D., 2012
86. 86. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70The 7s have the 1, 2, 3… pattern. © Joan A. Cotter, Ph.D., 2012
87. 87. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70The 7s have the 1, 2, 3… pattern. © Joan A. Cotter, Ph.D., 2012
88. 88. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70The 7s have the 1, 2, 3… pattern. © Joan A. Cotter, Ph.D., 2012
89. 89. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
90. 90. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
91. 91. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
92. 92. Multiples Memory © Joan A. Cotter, Ph.D., 2012
93. 93. Multiples MemoryObjective: To help the players learn themultiples patterns. © Joan A. Cotter, Ph.D., 2012
94. 94. Multiples MemoryObjective: To help the players learn themultiples patterns.Object of the game: To be the first player to collect all tencards of a multiple in order. © Joan A. Cotter, Ph.D., 2012
95. 95. Multiples Memory 7 14 21 28 35 42 49 56 63 70The 7s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
96. 96. Multiples Memory 8 16 24 32 40 48 56 64 72 80The 8s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
97. 97. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
98. 98. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
99. 99. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
100. 100. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 63 1470 © Joan A. Cotter, Ph.D., 2012
101. 101. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
102. 102. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
103. 103. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 40 © Joan A. Cotter, Ph.D., 2012
104. 104. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
105. 105. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
106. 106. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
107. 107. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
108. 108. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
109. 109. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
110. 110. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
111. 111. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 63 5670 8 © Joan A. Cotter, Ph.D., 2012
112. 112. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
113. 113. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
114. 114. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
115. 115. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 87 © Joan A. Cotter, Ph.D., 2012
116. 116. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 63 1470 87 © Joan A. Cotter, Ph.D., 2012
117. 117. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 87 14 © Joan A. Cotter, Ph.D., 2012
118. 118. Multiples Memory 7 14 21 28 35 42 49 56 63 70 24 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 87 14 © Joan A. Cotter, Ph.D., 2012
119. 119. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 87 14 © Joan A. Cotter, Ph.D., 2012
120. 120. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
121. 121. Multiplication MemoryObjective: To help the players master themultiplication facts. © Joan A. Cotter, Ph.D., 2012
122. 122. Multiplication MemoryObjective: To help the players master themultiplication facts.Object of the game: To collect the most cards by matchingthe multiplier with the product. © Joan A. Cotter, Ph.D., 2012
123. 123. Multiplication MemoryMaterials Needed: © Joan A. Cotter, Ph.D., 2012
124. 124. Multiplication Memory1 2 3 4 5 6 7 8 9 10 Materials Needed: • Ten basic cards, numbered 1 to 10 © Joan A. Cotter, Ph.D., 2012
125. 125. Multiplication Memory 31 2 3 4 5 3 6 9 12 15 18 6 7 8 9 10 21 24 27 30 Materials Needed: • Ten basic cards, numbered 1 to 10 • A set of product cards (3s used here) © Joan A. Cotter, Ph.D., 2012
126. 126. Multiplication Memory 31 2 3 4 5 3x 3 6 9 12 15 18 6 7 8 9 10 21 24 27 30 Materials Needed: • Ten basic cards, numbered 1 to 10 • A set of product cards (3s used here) • A stickie note with “3 x” written on it © Joan A. Cotter, Ph.D., 2012
127. 127. Multiplication Memory 31 2 3 4 5 3x 3 6 9 12 15 18 6 7 8 9 10 21 24 27 30 = Materials Needed: • Ten basic cards, numbered 1 to 10 • A set of product cards (3s used here) • A stickie with “3 x” written on it • A stickie with “=” written on it © Joan A. Cotter, Ph.D., 2012
128. 128. Multiplication Memory 31 2 3 4 5 3x 3 6 9 12 15 18 6 7 8 9 10 21 24 27 30 = Materials Needed: • Ten basic cards, numbered 1 to 10 • A set of product cards (3s used here) • A stickie with “3 x” written on it • A stickie with “=” written on it • A manipulative with groups of five © Joan A. Cotter, Ph.D., 2012
129. 129. Multiplication Memory3x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
130. 130. Multiplication Memory3x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
131. 131. Multiplication Memory 53x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
132. 132. Multiplication Memory 53x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
133. 133. Multiplication Memory 53x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
134. 134. Multiplication Memory 53x =3 taken 5 times equals 15. 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
135. 135. Multiplication Memory 5 213x =3 taken 5 times equals 15. 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
136. 136. Multiplication Memory3x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
137. 137. Multiplication Memory3x = 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
138. 138. Multiplication Memory3x = 7 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
139. 139. Multiplication Memory3x = 7 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
140. 140. Multiplication Memory3x = 7 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
141. 141. Multiplication Memory3x = 7 3 taken 7 times equals 21. 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
142. 142. Multiplication Memory 213x = 7 3 taken 7 times equals 21. 3 6 9 12 15 18 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
143. 143. Multiplication Memory3x = 3 taken 7 times equals 21. 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
144. 144. Multiplication Memory3x = 2 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
145. 145. Multiplication Memory3x = 2 3 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
146. 146. Multiplication Memory3x = 2 3 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
147. 147. Multiplication Memory3x = 2 3 3 taken 3 times equals 9. 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
148. 148. Multiplication Memory3x = 2 3 12 3 taken 3 times equals 9. 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
149. 149. Multiplication Memory3x = 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
150. 150. Multiplication Memory3x = 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
151. 151. Multiplication Memory 53x = 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
152. 152. Multiplication Memory 53x = 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
153. 153. Multiplication Memory 53x =3 taken 5 times equals 15. 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
154. 154. Multiplication Memory 53x = 153 taken 5 times equals 15. 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
155. 155. Multiplication Memory3x =3 taken 5 times equals 15. 5 15 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
156. 156. Multiplication Memory3x = 5 15 3 6 9 12 15 18 7 21 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
157. 157. Multiplication Memory3x = 8 24 3 6 9 12 15 18 1 3 21 24 27 30 © Joan A. Cotter, Ph.D., 2012
158. 158. Multiplication Tables © Joan A. Cotter, Ph.D., 2012
159. 159. Multiplication Tables © Joan A. Cotter, Ph.D., 2012
160. 160. Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
161. 161. Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
162. 162. Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
163. 163. Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
164. 164. Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
165. 165. Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
166. 166. Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
167. 167. Multiplication Tables © Joan A. Cotter, Ph.D., 2012
168. 168. Multiplication Tables 9×3 © Joan A. Cotter, Ph.D., 2012
169. 169. Multiplication Tables 9×3 © Joan A. Cotter, Ph.D., 2012
170. 170. Multiplication Tables 6×6 © Joan A. Cotter, Ph.D., 2012
171. 171. Multiplication Tables 6×6 © Joan A. Cotter, Ph.D., 2012
172. 172. Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
173. 173. Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
174. 174. Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
175. 175. Multiplication Tables 7×9 © Joan A. Cotter, Ph.D., 2012
176. 176. Multiplication Tables 7×9 © Joan A. Cotter, Ph.D., 2012
177. 177. Multiplication Tables 7×9 9×7 © Joan A. Cotter, Ph.D., 2012
178. 178. Multiplication Tables 7×9 9×7 © Joan A. Cotter, Ph.D., 2012
179. 179. Multiplication Tables squares © Joan A. Cotter, Ph.D., 2012
180. 180. Multiplication Tables squares © Joan A. Cotter, Ph.D., 2012
181. 181. Fraction Chart © Joan A. Cotter, Ph.D., 2012
182. 182. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10 © Joan A. Cotter, Ph.D., 2012
183. 183. Fraction ChartStairs (Unit fractions) 1 © Joan A. Cotter, Ph.D., 2012
184. 184. Fraction Chart Stairs (Unit fractions)12 1 © Joan A. Cotter, Ph.D., 2012
185. 185. Fraction Chart Stairs (Unit fractions)13 1 2 1 © Joan A. Cotter, Ph.D., 2012
186. 186. Fraction Chart Stairs (Unit fractions)14 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
187. 187. Fraction Chart Stairs (Unit fractions)15 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
188. 188. Fraction Chart Stairs (Unit fractions)16 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
189. 189. Fraction Chart Stairs (Unit fractions)17 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
190. 190. Fraction Chart Stairs (Unit fractions)18 1 7 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
191. 191. Fraction Chart Stairs (Unit fractions)1918 1 7 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
192. 192. Fraction Chart1 Stairs (Unit fractions)10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
193. 193. Fraction Chart1 Stairs (Unit fractions)10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
194. 194. Fraction Chart1 Stairs (Unit fractions)10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 A hyperbola. © Joan A. Cotter, Ph.D., 2012
195. 195. Fraction ChartStairs (Unit fractions) Without numbers. © Joan A. Cotter, Ph.D., 2012
196. 196. Fraction NamingOne divided into 2 equal parts. © Joan A. Cotter, Ph.D., 2012
197. 197. Fraction NamingOne divided into 2 equal parts.One divided into 3 equal parts. © Joan A. Cotter, Ph.D., 2012
198. 198. Fraction NamingOne divided into 2 equal parts.One divided into 3 equal parts.One divided into 4 equal parts. © Joan A. Cotter, Ph.D., 2012
199. 199. Fraction Naming 1One divided into 3 equal parts. 1 One whole © Joan A. Cotter, Ph.D., 2012
200. 200. Fraction Naming 1One divided into 3 equal parts. 1 One whole divided into © Joan A. Cotter, Ph.D., 2012
201. 201. Fraction Naming 1One divided into 3 equal parts. 1 One whole divided into 3 three equal parts © Joan A. Cotter, Ph.D., 2012
202. 202. Fraction Naming 1One divided into 3 equal parts. 1 One divided by 3 three © Joan A. Cotter, Ph.D., 2012
203. 203. Fraction Naming 1 One divided into 3 equal parts. 1 One divided by 3 threeIn English, except for half, we use ordinal numbersto name fractions. © Joan A. Cotter, Ph.D., 2012
204. 204. Fraction Naming 1One divided into 3 equal parts. 1 Read as “one-third.” 3 © Joan A. Cotter, Ph.D., 2012
205. 205. Fraction Naming 11 1 13 3 3 One divided into 3 equal parts. 1 Read as “one-third.” 3 © Joan A. Cotter, Ph.D., 2012
206. 206. Unit Fraction WarObjective: To help the children realize a unit fractiondecreases as the denominator increases. © Joan A. Cotter, Ph.D., 2012
207. 207. Unit Fraction WarObjective: To help the children realize a unit fractiondecreases as the denominator increases.Object of the game: To collect all, or most, of the cards withthe greater unit fraction. © Joan A. Cotter, Ph.D., 2012
208. 208. Unit Fraction War © Joan A. Cotter, Ph.D., 2012
209. 209. Unit Fraction War1 14 5 © Joan A. Cotter, Ph.D., 2012
210. 210. Unit Fraction War1 14 5 © Joan A. Cotter, Ph.D., 2012
211. 211. Unit Fraction War © Joan A. Cotter, Ph.D., 2012
212. 212. Unit Fraction War1 18 © Joan A. Cotter, Ph.D., 2012
213. 213. Unit Fraction War1 18 © Joan A. Cotter, Ph.D., 2012
214. 214. Unit Fraction War © Joan A. Cotter, Ph.D., 2012
215. 215. Unit Fraction War1 16 6 © Joan A. Cotter, Ph.D., 2012
216. 216. Unit Fraction War1 16 6 © Joan A. Cotter, Ph.D., 2012
217. 217. Unit Fraction War1 16 6 1 1 3 4 © Joan A. Cotter, Ph.D., 2012
218. 218. Unit Fraction War © Joan A. Cotter, Ph.D., 2012
219. 219. Fraction Naming 11 1 13 3 3 © Joan A. Cotter, Ph.D., 2012
220. 220. Fraction Naming 11 1 13 3 3 © Joan A. Cotter, Ph.D., 2012
221. 221. Fraction Naming 11 1 13 3 3 © Joan A. Cotter, Ph.D., 2012
222. 222. Fraction Naming 11 1 13 3 3 Two 1s 3 © Joan A. Cotter, Ph.D., 2012
223. 223. Fraction Naming 11 1 13 3 3 Two 1s is 2 3 3 © Joan A. Cotter, Ph.D., 2012
224. 224. Fraction Naming 11 1 13 3 3 Two 1s is 2 3 3 Read as “two-thirds.” © Joan A. Cotter, Ph.D., 2012
225. 225. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many fourths in a whole? © Joan A. Cotter, Ph.D., 2012
226. 226. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many fourths in a whole? © Joan A. Cotter, Ph.D., 2012
227. 227. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many fourths in a whole? © Joan A. Cotter, Ph.D., 2012
228. 228. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many fourths in a whole? © Joan A. Cotter, Ph.D., 2012
229. 229. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many fourths in a whole? © Joan A. Cotter, Ph.D., 2012
230. 230. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10How many eighths in a whole? © Joan A. Cotter, Ph.D., 2012
231. 231. Concentrating on One GameObjective: To help the children realize that 5 fifths, 8eighths, and so forth, make a whole. © Joan A. Cotter, Ph.D., 2012
232. 232. Concentrating on One GameObjective: To help the children realize that 5 fifths, 8eighths, and so forth, make a whole.Object of the game: To find the pairs that make a whole. © Joan A. Cotter, Ph.D., 2012
233. 233. Concentrating on One © Joan A. Cotter, Ph.D., 2012
234. 234. Concentrating on One 3 5 © Joan A. Cotter, Ph.D., 2012
235. 235. Concentrating on One 3 5 © Joan A. Cotter, Ph.D., 2012
236. 236. Concentrating on One 3 5 © Joan A. Cotter, Ph.D., 2012
237. 237. Concentrating on One 3 5 2 5 © Joan A. Cotter, Ph.D., 2012
238. 238. Concentrating on One © Joan A. Cotter, Ph.D., 2012
239. 239. Concentrating on One 3 8 © Joan A. Cotter, Ph.D., 2012
240. 240. Concentrating on One 3 8 © Joan A. Cotter, Ph.D., 2012
241. 241. Concentrating on One 3 8 © Joan A. Cotter, Ph.D., 2012
242. 242. Concentrating on One 7 8 3 8 © Joan A. Cotter, Ph.D., 2012
243. 243. Concentrating on One © Joan A. Cotter, Ph.D., 2012
244. 244. Concentrating on One 5 8 © Joan A. Cotter, Ph.D., 2012
245. 245. Concentrating on One 5 8 © Joan A. Cotter, Ph.D., 2012
246. 246. Concentrating on One 5 8 © Joan A. Cotter, Ph.D., 2012
247. 247. Concentrating on One 5 8 3 8 © Joan A. Cotter, Ph.D., 2012
248. 248. Concentrating on One © Joan A. Cotter, Ph.D., 2012
249. 249. Concentrating on One © Joan A. Cotter, Ph.D., 2012
250. 250. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
251. 251. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
252. 252. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 7/8 or 8/9? © Joan A. Cotter, Ph.D., 2012
253. 253. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 7/8 or 8/9? © Joan A. Cotter, Ph.D., 2012
254. 254. Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10An interesting pattern. © Joan A. Cotter, Ph.D., 2012
255. 255. Partial Chart 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
256. 256. Partial Chart 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
257. 257. Partial Chart © Joan A. Cotter, Ph.D., 2012
258. 258. Partial Chart1 2 3 4 5 6 © Joan A. Cotter, Ph.D., 2012
259. 259. Fraction WarObjective: To practice comparing ones, halves,fourths, and eighths in preparation forreading a ruler. © Joan A. Cotter, Ph.D., 2012
260. 260. Fraction WarObjective: To practice comparing ones, halves,fourths, and eighths in preparation forreading a ruler.Object of the game: To capture all the cards. © Joan A. Cotter, Ph.D., 2012
261. 261. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
262. 262. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 81 18 4 © Joan A. Cotter, Ph.D., 2012
263. 263. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 81 18 4 © Joan A. Cotter, Ph.D., 2012
264. 264. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 81 18 4 © Joan A. Cotter, Ph.D., 2012
265. 265. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
266. 266. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 85 38 4 © Joan A. Cotter, Ph.D., 2012
267. 267. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 85 38 4 © Joan A. Cotter, Ph.D., 2012
268. 268. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 85 38 4 © Joan A. Cotter, Ph.D., 2012
269. 269. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
270. 270. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 83 34 4 © Joan A. Cotter, Ph.D., 2012
271. 271. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 83 34 4 © Joan A. Cotter, Ph.D., 2012
272. 272. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 83 34 4 3 1 8 4 © Joan A. Cotter, Ph.D., 2012
273. 273. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
274. 274. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 41 1 1 1 1 1 1 18 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
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