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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