IMF: Visualization October 2011

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  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Stairs
  • IMF: Visualization October 2011

    1. 1. Adding Visualization to Montessori Mathematics IMF Conference October 21, 2011 Sarasota, Florida by Joan A. Cotter, Ph.D. [email_address] Presentation available: ALabacus.com 7 x 7 VII 1000 10 1 100 7 3 7 3
    2. 2. Counting Model <ul><li>Number Rods </li></ul><ul><li>Spindle Boxes </li></ul><ul><li>Golden Bead materials </li></ul><ul><li>Snake Game </li></ul><ul><li>Dot Game </li></ul><ul><li>Stamp Game </li></ul><ul><li>Multiplication Board </li></ul><ul><li>Bead Frame </li></ul>In Montessori, counting is pervasive:
    3. 3. Counting Model From a child's perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth
    4. 4. Counting Model From a child's perspective F + E
    5. 5. Counting Model From a child's perspective F + E A
    6. 6. Counting Model From a child's perspective F + E A B
    7. 7. Counting Model From a child's perspective F + E A C B
    8. 8. Counting Model From a child's perspective F + E A F C D E B
    9. 9. Counting Model From a child's perspective F + E A A F C D E B
    10. 10. Counting Model From a child's perspective F + E A B A F C D E B
    11. 11. Counting Model From a child's perspective F + E A C D E B A F C D E B
    12. 12. Counting Model From a child's perspective F + E What is the sum? (It must be a letter.) A C D E B A F C D E B
    13. 13. Counting Model From a child's perspective K G I J K H A F C D E B F + E
    14. 14. Counting Model From a child's perspective Now memorize the facts!! G + D
    15. 15. Counting Model From a child's perspective Now memorize the facts!! G + D H + F
    16. 16. Counting Model From a child's perspective Now memorize the facts!! G + D H + F D + C
    17. 17. Counting Model From a child's perspective Now memorize the facts!! G + D H + F C + G D + C
    18. 18. Counting Model From a child's perspective Now memorize the facts!! E + I G + D H + F C + G D + C
    19. 19. Counting Model From a child's perspective Try subtracting by “taking away” H – E
    20. 20. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T .
    21. 21. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T . What is D  E ?
    22. 22. Counting Model From a child's perspective L is written AB because it is A J and B A’s
    23. 23. Counting Model From a child's perspective L is written AB because it is A J and B A’s huh?
    24. 24. Counting Model From a child's perspective L is written AB because it is A J and B A’s (twelve)
    25. 25. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (twelve)
    26. 26. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (twelve)
    27. 27. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (two 1s). (twelve)
    28. 28. Counting Model Summary
    29. 29. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul>Summary
    30. 30. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul>Summary
    31. 31. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul>Summary
    32. 32. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul>Summary
    33. 33. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul><ul><li>Is tedious and time-consuming. </li></ul>Summary
    34. 34. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul><ul><li>Is tedious and time-consuming. </li></ul>Summary <ul><li>Does not provide an efficient way to master the facts. </li></ul>
    35. 35. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
    36. 36. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
    37. 37. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31
    38. 38. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.
    39. 39. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3.
    40. 40. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
    41. 41. Calendar Math August 8 1 9 2 10 3 4 5 6 7 Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.
    42. 42. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul>
    43. 43. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul><ul><li>Children need to see the whole month, not just part. </li></ul><ul><ul><li>Purpose of calendar is to plan ahead. </li></ul></ul><ul><ul><li>Many ways to show the current date. </li></ul></ul>
    44. 44. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul><ul><li>Children need to see the whole month, not just part. </li></ul><ul><ul><li>Purpose of calendar is to plan ahead. </li></ul></ul><ul><ul><li>Many ways to show the current date. </li></ul></ul><ul><li>Calendars give a narrow view of patterning. </li></ul><ul><ul><li>Patterns do not necessarily involve numbers. </li></ul></ul><ul><ul><li>Patterns rarely proceed row by row. </li></ul></ul><ul><ul><li>Patterns go on forever; they don’t stop at 31. </li></ul></ul>
    45. 45. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    46. 46. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    47. 47. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    48. 48. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    49. 49. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    50. 50. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    51. 51. Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    52. 52. Memorizing Math Flash cards: 9 + 7
    53. 53. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul>9 + 7
    54. 54. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul>9 + 7
    55. 55. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Don’t work for those with learning disabilities. </li></ul>9 + 7
    56. 56. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Don’t work for those with learning disabilities. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul>9 + 7
    57. 57. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Don’t work for those with learning disabilities. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul><ul><li>Often produce stress – children under stress stop learning. </li></ul>9 + 7
    58. 58. Memorizing Math Flash cards: <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Don’t work for those with learning disabilities. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul><ul><li>Often produce stress – children under stress stop learning. </li></ul><ul><li>Are not concrete – use abstract symbols. </li></ul>9 + 7
    59. 59. Research on Counting Karen Wynn’s research Show the baby two teddy bears.
    60. 60. Research on Counting Karen Wynn’s research Then hide them with a screen.
    61. 61. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.
    62. 62. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.
    63. 63. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.
    64. 64. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.
    65. 65. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.
    66. 66. Research on Counting <ul><li>Other research </li></ul>
    67. 67. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul>Other research These groups matched quantities without using counting words.
    68. 68. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul>Other research These groups matched quantities without using counting words.
    69. 69. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Adults, ages 18-50, from Boston. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul>Other research These groups matched quantities without using counting words.
    70. 70. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Adults, ages 18-50, from Boston. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Baby chicks from Italy. </li></ul><ul><ul><li>Lucia Regolin, University of Padova, 2009. </li></ul></ul>Other research These groups matched quantities without using counting words.
    71. 71. Research on Counting In Japanese schools: <ul><li>Children are discouraged from using counting for adding. </li></ul>
    72. 72. Research on Counting In Japanese schools: <ul><li>Children are discouraged from using counting for adding. </li></ul><ul><li>They consistently group in 5s. </li></ul>
    73. 73. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul>
    74. 74. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul><ul><li>Human babies and some animals can subitize small quantities at birth. </li></ul>
    75. 75. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul><ul><li>“ One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”— Montessori </li></ul><ul><li>Human babies and some animals can subitize small quantities at birth. </li></ul>Said to contrast to teaching the names of a triangle and square by counting.
    76. 76. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul><ul><li>“ One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”— Montessori </li></ul><ul><li>Subitizing “allows the child to grasp the whole and the elements at the same time.”— Benoit </li></ul><ul><li>Human babies and some animals can subitize small quantities at birth. </li></ul>
    77. 77. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul><ul><li>“ One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”— Montessori </li></ul><ul><li>Subitizing “allows the child to grasp the whole and the elements at the same time.”— Benoit </li></ul><ul><li>Subitizing seems to be a necessary skill for under-standing what the counting process means.— Glasersfeld </li></ul><ul><li>Human babies and some animals can subitize small quantities at birth. </li></ul>
    78. 78. Research on Counting Subitizing <ul><li>Subitizing is recognizing a quantity without counting. </li></ul><ul><li>“ One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”— Montessori </li></ul><ul><li>Children who can subitize perform better in mathematics.— Butterworth </li></ul><ul><li>Subitizing “allows the child to grasp the whole and the elements at the same time.”— Benoit </li></ul><ul><li>Subitizing seems to be a necessary skill for under-standing what the counting process means.— Glasersfeld </li></ul><ul><li>Subitizing not affected by math anxiety; counting ability is affected. </li></ul><ul><li>Human babies and some animals can subitize small quantities at birth. </li></ul>
    79. 79. Research on Counting Finger gnosia <ul><li>Finger gnosia is the ability to know which fingers can been lightly touched without looking. </li></ul>
    80. 80. Research on Counting Finger gnosia <ul><li>Finger gnosia is the ability to know which fingers can been lightly touched without looking. </li></ul><ul><li>Part of the brain controlling fingers is adjacent to math part of the brain. </li></ul>
    81. 81. Research on Counting Finger gnosia <ul><li>Finger gnosia is the ability to know which fingers can been lightly touched without looking. </li></ul><ul><li>Part of the brain controlling fingers is adjacent to math part of the brain. </li></ul><ul><li>Children who use their fingers as representational tools perform better in mathematics— Butterworth </li></ul>
    82. 82. Visualizing Mathematics
    83. 83. Visualizing Mathematics “ In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E I)
    84. 84. Visualizing Mathematics “ Think in pictures, because the brain remembers images better than it does anything else.”   Ben Pridmore, World Memory Champion, 2009
    85. 85. Visualizing Mathematics “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb
    86. 86. Visualizing Mathematics “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
    87. 87. Visualizing Mathematics “ The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others
    88. 88. Visualizing Mathematics Japanese criteria for manipulatives Japanese Council of Mathematics Education <ul><li>Representative of structure of numbers. </li></ul><ul><li>Easily manipulated by children. </li></ul><ul><li>Imaginable mentally. </li></ul>
    89. 89. Visualizing Mathematics Visualizing also needed in: <ul><li>Reading </li></ul><ul><li>Sports </li></ul><ul><li>Creativity </li></ul><ul><li>Geography </li></ul><ul><li>Engineering </li></ul><ul><li>Construction </li></ul>
    90. 90. Visualizing Mathematics Visualizing also needed in: <ul><li>Reading </li></ul><ul><li>Sports </li></ul><ul><li>Creativity </li></ul><ul><li>Geography </li></ul><ul><li>Engineering </li></ul><ul><li>Construction </li></ul><ul><li>Architecture </li></ul><ul><li>Astronomy </li></ul><ul><li>Archeology </li></ul><ul><li>Chemistry </li></ul><ul><li>Physics </li></ul><ul><li>Surgery </li></ul>
    91. 91. Visualizing Mathematics Ready: How many?
    92. 92. Visualizing Mathematics Ready: How many?
    93. 93. Visualizing Mathematics Try again: How many?
    94. 94. Visualizing Mathematics Try again: How many?
    95. 95. Visualizing Mathematics Try again: How many?
    96. 96. Visualizing Mathematics Ready: How many?
    97. 97. Visualizing Mathematics Try again: How many?
    98. 98. Visualizing Mathematics Try to visualize 8 identical apples without grouping.
    99. 99. Visualizing Mathematics Try to visualize 8 identical apples without grouping.
    100. 100. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
    101. 101. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
    102. 102. Visualizing Mathematics Early Roman numerals I II III IIII V VIII 1 2 3 4 5 8 Romans grouped in fives. Notice 8 is 5 and 3.
    103. 103. Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.
    104. 104. Naming Quantities Using fingers
    105. 105. Naming Quantities Using fingers Naming quantities is a three-period lesson.
    106. 106. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.
    107. 107. Naming Quantities Using fingers
    108. 108. Naming Quantities Using fingers
    109. 109. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.
    110. 110. Naming Quantities Using fingers
    111. 111. Naming Quantities Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. – Joan A. Cotter Also set to music. Listen and download sheet music from Web site.
    112. 112. Naming Quantities Recognizing 5
    113. 113. Naming Quantities Recognizing 5
    114. 114. Naming Quantities Recognizing 5 5 has a middle; 4 does not. Look at your hand; your middle finger is longer to remind you 5 has a middle.
    115. 115. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.
    116. 116. Naming Quantities Tally sticks
    117. 117. Naming Quantities Tally sticks
    118. 118. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.
    119. 119. Naming Quantities Tally sticks
    120. 120. Naming Quantities Tally sticks Start a new row for every ten.
    121. 121. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?
    122. 122. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.
    123. 123. Naming Quantities Number Control Chart 1 2 3 4 5 To help the child learn the symbols
    124. 124. Naming Quantities Number Control Chart 6 1 7 2 8 3 9 4 10 5 To help the child learn the symbols
    125. 125. Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.
    126. 126. Naming Quantities Ordering Finger Cards Putting the finger cards in order.
    127. 127. Naming Quantities 10 Matching Numbers to Finger Cards Match the number to the finger card. 5 1
    128. 128. Naming Quantities Matching Fingers to Number Cards Match the finger card to the number. 9 4 1 6 10 2 8 3 5 7
    129. 129. Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.
    130. 130. Naming Quantities Number Rods
    131. 131. Naming Quantities Number Rods
    132. 132. Naming Quantities Number Rods Using different colors.
    133. 133. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate containers.
    134. 134. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.
    135. 135. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 1 2 3 0 4
    136. 136. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    137. 137. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    138. 138. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    139. 139. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    140. 140. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    141. 141. Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right. 6 7 8 5 9
    142. 142. Naming Quantities “ Grouped in fives so the child does not need to count.” Black and White Bead Stairs A. M. Joosten This was the inspiration to group in 5s.
    143. 143. AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Trading Side introduces algorithms with trading. 1000 10 1 100
    144. 144. AL Abacus Cleared
    145. 145. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.
    146. 146. 5 AL Abacus Entering quantities Relate quantities to hands.
    147. 147. 7 AL Abacus Entering quantities
    148. 148. AL Abacus 10 Entering quantities
    149. 149. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.
    150. 150. AL Abacus Adding
    151. 151. AL Abacus Adding 4 + 3 =
    152. 152. AL Abacus Adding 4 + 3 =
    153. 153. AL Abacus Adding 4 + 3 =
    154. 154. AL Abacus Adding 4 + 3 =
    155. 155. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.
    156. 156. Problem Solving Part-Whole Circles Part-whole circles children see relationships and solve problems.
    157. 157. Problem Solving Part-Whole Circles Whole Part-whole circles children see relationships and solve problems.
    158. 158. Problem Solving Part-Whole Circles Whole Part-whole circles children see relationships and solve problems. Part Part
    159. 159. Problem Solving Part-Whole Circles 10 If 10 is the whole
    160. 160. Problem Solving Part-Whole Circles 10 4 and 4 is one part,
    161. 161. Problem Solving Part-Whole Circles 10 4 What is the other part?
    162. 162. Problem Solving Part-Whole Circles 10 4 6 What is the other part?
    163. 163. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.
    164. 164. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole?
    165. 165. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole? 3
    166. 166. Problem Solving Solving a problem 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole?
    167. 167. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole ? 5 3
    168. 168. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 5 3 What is the missing part?
    169. 169. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part? 5 3 2
    170. 170. Problem Solving Solving a problem 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Write the equation.
    171. 171. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 Write the equation.
    172. 172. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 Write the equation.
    173. 173. Problem Solving Solving a problem Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 5 – 3 = 2 Write the equation. Is this an addition or subtraction problem?
    174. 174. Problem Solving Part-whole circles young children solve problems. Writing equations do not.
    175. 175. Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Children use the abacus while playing this “Go Fish” type game.
    176. 176. Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Object of the game: To collect the most pairs that equal ten. Children use the abacus while playing this “Go Fish” type game.
    177. 177. Go to the Dump Game The ways to partition 10.
    178. 178. Go to the Dump Game A game viewed from above. Starting
    179. 179. Go to the Dump Game Each player takes 5 cards. 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9 Starting
    180. 180. Go to the Dump Game Does YellowCap have any pairs? [no] Finding pairs 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9
    181. 181. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] Finding pairs 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9
    182. 182. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] Finding pairs 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9
    183. 183. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6
    184. 184. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6
    185. 185. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6
    186. 186. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] Finding pairs 7 2 7 9 5 2 1 8 3 4 9 4 6 7 3
    187. 187. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] Finding pairs 7 2 7 9 5 1 3 4 9 4 6 2 8 2 8
    188. 188. Go to the Dump Game The player asks the player on her left. Playing 7 2 7 9 5 1 3 4 9 4 6 2 8 2 8
    189. 189. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. Playing 7 2 7 9 5 1 3 4 9 4 6 2 8 2 8
    190. 190. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. Playing 7 2 7 9 5 1 3 4 9 4 6 2 8 2 8
    191. 191. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    192. 192. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? YellowCap gets another turn. Playing 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    193. 193. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. YellowCap gets another turn. Playing 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    194. 194. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. 2 Playing 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    195. 195. Go to the Dump Game Playing 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    196. 196. Go to the Dump Game PinkCap, do you have a 6? Playing 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    197. 197. Go to the Dump Game PinkCap, do you have a 6? Go to the dump. Playing 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    198. 198. Go to the Dump Game 5 Playing 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    199. 199. Go to the Dump Game Playing 5 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    200. 200. Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 9 5 1 4 9 4 6 2 8 2 8 7 3
    201. 201. Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 6 2 8 2 8 7 3
    202. 202. Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 6 2 8 2 8 7 3 9
    203. 203. Go to the Dump Game Playing 5 2 2 7 5 4 9 4 6 2 8 1 9 7 3
    204. 204. Go to the Dump Game PinkCap is not out of the game. Her turn ends, but she takes 5 more cards. 2 9 1 7 7 Playing 5 2 2 7 5 4 9 4 6 2 8 1 9 7 3
    205. 205. Go to the Dump Game Winner? 5 5 4 6 9 1
    206. 206. Go to the Dump Game No counting. Combine both stacks. Winner? 5 5 4 6 9 1
    207. 207. Go to the Dump Game Whose stack is the highest? Winner? 5 5 4 6 9 1
    208. 208. Go to the Dump Game No shuffling needed for next game. Next game
    209. 209. “ Math” Way of Naming Numbers
    210. 210. “ Math” Way of Naming Numbers 11 = ten 1
    211. 211. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2
    212. 212. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3
    213. 213. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
    214. 214. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
    215. 215. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    216. 216. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    217. 217. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    218. 218. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    219. 219. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
    220. 220. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.
    221. 221. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 Only numbers under 100 need to be said the “math” way.
    222. 222. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Shows how far children from 3 countries can count at ages 4, 5, and 6.
    223. 223. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Purple is Chinese. Note jump between ages 5 and 6.
    224. 224. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Dark green is Korean “math” way.
    225. 225. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.
    226. 226. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Red is English speakers. They learn same amount between ages 4-5 and 5-6.
    227. 227. Math Way of Naming Numbers <ul><li>Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) </li></ul>
    228. 228. Math Way of Naming Numbers <ul><li>Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul>
    229. 229. Math Way of Naming Numbers <ul><li>Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul><ul><li>They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. </li></ul>
    230. 230. Math Way of Naming Numbers <ul><li>Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul><ul><li>They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. </li></ul><ul><li>Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. </li></ul>
    231. 231. Math Way of Naming Numbers Compared to reading:
    232. 232. Math Way of Naming Numbers <ul><li>Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. </li></ul>Compared to reading:
    233. 233. Math Way of Naming Numbers <ul><li>Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. </li></ul><ul><li>Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). </li></ul>Compared to reading:
    234. 234. Math Way of Naming Numbers <ul><li>Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. </li></ul><ul><li>Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). </li></ul><ul><li>Montessorians do use the math way of naming numbers but are too quick to switch to traditional names. Use the math way for a longer period of time. </li></ul>Compared to reading:
    235. 235. Math Way of Naming Numbers “ Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers
    236. 236. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:
    237. 237. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.
    238. 238. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.
    239. 239. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    240. 240. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    241. 241. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    242. 242. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    243. 243. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    244. 244. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    245. 245. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    246. 246. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    247. 247. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    248. 248. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    249. 249. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.
    250. 250. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. The traditional names for 40, 60, 70, 80, and 90 follow a pattern.
    251. 251. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.
    252. 252. Math Way of Naming Numbers Traditional names 3-ten = thirty “ Thir” also used in 1/3, 13 and 30.
    253. 253. Math Way of Naming Numbers Traditional names 5-ten = fifty “ Fif” also used in 1/5, 15 and 50.
    254. 254. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”
    255. 255. Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.
    256. 256. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.
    257. 257. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.
    258. 258. Math Way of Naming Numbers Traditional names ten 4 “ Teen” also means ten.
    259. 259. Math Way of Naming Numbers Traditional names ten 4 teen 4 “ Teen” also means ten.
    260. 260. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “ Teen” also means ten.
    261. 261. Math Way of Naming Numbers Traditional names a one left
    262. 262. Math Way of Naming Numbers Traditional names a one left a left-one
    263. 263. Math Way of Naming Numbers Traditional names a one left a left-one eleven
    264. 264. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”
    265. 265. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”
    266. 266. Composing Numbers 3-ten
    267. 267. Composing Numbers 3-ten
    268. 268. Composing Numbers 3-ten 3 0
    269. 269. Composing Numbers 3-ten 3 0 Point to the 3 and say 3.
    270. 270. Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.
    271. 271. Composing Numbers 3-ten 7 3 0
    272. 272. Composing Numbers 3-ten 7 3 0
    273. 273. Composing Numbers 3-ten 7 3 0 7
    274. 274. Composing Numbers 3-ten 7 3 0 7 Place the 7 on top of the 0 of the 30.
    275. 275. Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 0 7
    276. 276. Composing Numbers 7-ten 8 7 8 8 Another example.
    277. 277. Composing Numbers 10-ten
    278. 278. Composing Numbers 10-ten 1 0 0
    279. 279. Composing Numbers 10-ten 1 0 0
    280. 280. Composing Numbers 10-ten 1 0 0
    281. 281. Composing Numbers 1 hundred
    282. 282. Composing Numbers 1 hundred 1 0 0
    283. 283. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.
    284. 284. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred. 1 0 1 0
    285. 285. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.
    286. 286. Composing Numbers Reading numbers backward 2 5 8 4 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
    287. 287. Composing Numbers 2 5 8 4 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    288. 288. Composing Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    289. 289. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    290. 290. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward The Decimal Cards encourage reading numbers in the normal order.
    291. 291. Composing Numbers In scientific notation, we “stand” on the left digit and note the number of digits to the right. (That’s why we shouldn’t refer to the 4 as the 4th column.) Scientific Notation 4000 = 4 x 10 3
    292. 292. Fact Strategies
    293. 293. Fact Strategies <ul><li>A strategy is a way to learn a new fact or recall a forgotten fact. </li></ul>
    294. 294. Fact Strategies <ul><li>A strategy is a way to learn a new fact or recall a forgotten fact. </li></ul><ul><li>A visualizable representation is part of a powerful strategy. </li></ul>
    295. 295. Fact Strategies Complete the Ten 9 + 5 =
    296. 296. Fact Strategies Complete the Ten 9 + 5 =
    297. 297. Fact Strategies Complete the Ten 9 + 5 =
    298. 298. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
    299. 299. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the beads simultaneously.
    300. 300. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
    301. 301. Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
    302. 302. Fact Strategies Two Fives 8 + 6 =
    303. 303. Fact Strategies Two Fives 8 + 6 =
    304. 304. Fact Strategies Two Fives 8 + 6 = Two fives make 10.
    305. 305. Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”
    306. 306. Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”
    307. 307. Fact Strategies Two Fives 7 + 5 = Another example.
    308. 308. Fact Strategies Two Fives 7 + 5 =
    309. 309. Fact Strategies Two Fives 7 + 5 = 12
    310. 310. Fact Strategies Going Down 15 – 9 =
    311. 311. Fact Strategies Going Down 15 – 9 =
    312. 312. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    313. 313. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    314. 314. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    315. 315. Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
    316. 316. Fact Strategies Subtract from 10 15 – 9 =
    317. 317. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    318. 318. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    319. 319. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    320. 320. Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
    321. 321. Fact Strategies Going Up 13 – 9 =
    322. 322. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    323. 323. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    324. 324. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    325. 325. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    326. 326. Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.
    327. 327. Money Penny
    328. 328. Money Nickel
    329. 329. Money Dime
    330. 330. Money Quarter
    331. 331. Money Quarter
    332. 332. Money Quarter
    333. 333. Money Quarter
    334. 334. Bead Frame 1 10 100 1000
    335. 335. Bead Frame 8 + 6 1 10 100 1000
    336. 336. Bead Frame 8 + 6 1 10 100 1000
    337. 337. Bead Frame 8 + 6 1 10 100 1000
    338. 338. Bead Frame 8 + 6 1 10 100 1000
    339. 339. Bead Frame 8 + 6 1 10 100 1000
    340. 340. Bead Frame 8 + 6 1 10 100 1000
    341. 341. Bead Frame 8 + 6 1 10 100 1000
    342. 342. Bead Frame 8 + 6 1 10 100 1000
    343. 343. Bead Frame 8 + 6 1 10 100 1000
    344. 344. Bead Frame 8 + 6 14 1 10 100 1000
    345. 345. Bead Frame Difficulties for the child
    346. 346. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul>Difficulties for the child
    347. 347. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul><ul><li>Not visualizable: Beads need to be grouped in fives. </li></ul>Difficulties for the child
    348. 348. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul><ul><li>Not visualizable: Beads need to be grouped in fives. </li></ul><ul><li>Inconsistent with equation order when beads are moved right: Beads need to be moved left. </li></ul>Difficulties for the child
    349. 349. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul><ul><li>Not visualizable: Beads need to be grouped in fives. </li></ul><ul><li>Inconsistent with equation order when beads are moved right: Beads need to be moved left. </li></ul><ul><li>Hierarchies of numbers represented sideways: They need to be in vertical columns. </li></ul>Difficulties for the child
    350. 350. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul><ul><li>Not visualizable: Beads need to be grouped in fives. </li></ul><ul><li>Inconsistent with equation order when beads are moved right: Beads need to be moved left. </li></ul><ul><li>Hierarchies of numbers represented sideways: They need to be in vertical columns. </li></ul><ul><li>Trading done before second number is completely added: Addends need to combined before trading. </li></ul>Difficulties for the child
    351. 351. Bead Frame <ul><li>Distracting: Room is visible through the frame. </li></ul><ul><li>Not visualizable: Beads need to be grouped in fives. </li></ul><ul><li>Inconsistent with equation order when beads are moved right: Beads need to be moved left. </li></ul><ul><li>Hierarchies of numbers represented sideways: They need to be in vertical columns. </li></ul><ul><li>Trading done before second number is completely added: Addends need to combined before trading. </li></ul><ul><li>Answer is read going up: We read top to bottom. </li></ul>Difficulties for the child
    352. 352. Trading Side Cleared 1000 10 1 100
    353. 353. Trading Side Thousands 1000 10 1 100
    354. 354. Trading Side Hundreds 1000 10 1 100 The third wire from each end is not used.
    355. 355. Trading Side Tens 1000 10 1 100 The third wire from each end is not used.
    356. 356. Trading Side Ones 1000 10 1 100 The third wire from each end is not used.
    357. 357. Trading Side Adding 8 + 6 1000 10 1 100
    358. 358. Trading Side Adding 8 + 6 1000 10 1 100
    359. 359. Trading Side Adding 8 + 6 1000 10 1 100
    360. 360. Trading Side Adding 8 + 6 1000 10 1 100
    361. 361. Trading Side Adding 8 + 6 14 1000 10 1 100
    362. 362. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow). 1000 10 1 100
    363. 363. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100
    364. 364. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100
    365. 365. Trading Side Adding 8 + 6 14 Same answer before and after trading. 1000 10 1 100
    366. 366. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 1000 10 1 100
    367. 367. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100
    368. 368. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100
    369. 369. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    370. 370. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    371. 371. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    372. 372. Trading Side Bead Trading game 6 Trade 10 ones for 1 ten. 1000 10 1 100
    373. 373. Trading Side Bead Trading game 6 1000 10 1 100
    374. 374. Trading Side Bead Trading game 6 1000 10 1 100
    375. 375. Trading Side Bead Trading game 9 1000 10 1 100
    376. 376. Trading Side Bead Trading game 9 1000 10 1 100
    377. 377. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100
    378. 378. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100
    379. 379. Trading Side Bead Trading game 3 1000 10 1 100
    380. 380. Trading Side Bead Trading game 3 1000 10 1 100
    381. 381. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul>
    382. 382. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul>
    383. 383. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul>
    384. 384. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul><ul><li>Bead trading helps the child experience the greater value of each column from left to right. </li></ul>
    385. 385. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul><ul><li>Bead trading helps the child experience the greater value of each column from left to right. </li></ul><ul><li>To detect a pattern, there must be at least three examples in the sequence. (Place value is a pattern.) </li></ul>
    386. 386. Trading Side Adding 4-digit numbers 3658 + 2738 1000 10 1 100
    387. 387. Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right. 1000 10 1 100
    388. 388. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
    389. 389. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
    390. 390. Trading Side Adding 4-digit numbers 3 6 58 + 2738 Enter the first number from left to right. 1000 10 1 100
    391. 391. Trading Side Adding 4-digit numbers 36 5 8 + 2738 Enter the first number from left to right. 1000 10 1 100
    392. 392. Trading Side Adding 4-digit numbers 365 8 + 2738 Enter the first number from left to right. 1000 10 1 100
    393. 393. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    394. 394. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    395. 395. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    396. 396. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    397. 397. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen? 1000 10 1 100
    398. 398. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column? 1000 10 1 100
    399. 399. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 1000 10 1 100
    400. 400. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no] 1000 10 1 100
    401. 401. Trading Side Adding 4-digit numbers 3658 + 2738 9 6 Add starting at the right. Write results after each step. 1 1000 10 1 100
    402. 402. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    403. 403. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes] 1000 10 1 100
    404. 404. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3. 1000 10 1 100
    405. 405. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    406. 406. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    407. 407. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    408. 408. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    409. 409. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    410. 410. Trading Side Adding 4-digit numbers 3658 + 2738 6 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    411. 411. Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    412. 412. The Stamp Game 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1 1000 100 10 1 1000 100 10 1
    413. 413. The Stamp Game 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1 1000 100 10 1 1000 100 10 1
    414. 414. The Stamp Game 100 10 1 100 10 1 100 10 1 100 10 1 10 1 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 10 10 100 100 100 100 100 100 100 100
    415. 415. The Stamp Game 100 10 1 100 10 1 100 10 1 100 10 1 10 1 1 1000 100 10 1 1000 100 1 1000 100 10 1 1000 100 10 1 10 10 100 100 100 100 100 100 100 100 10
    416. 416. The Stamp Game 100 10 1 100 10 1 100 10 1 100 10 1 10 1 1 1000 100 10 1 1000 100 1 1000 100 10 1 1000 100 10 1 10 10 100 100 100 100 100 100 100 100 10
    417. 417. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    418. 418. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    419. 419. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    420. 420. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    421. 421. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    422. 422. Multiplication on the AL Abacus Basic facts 9  3 =
    423. 423. Multiplication on the AL Abacus Basic facts 9  3 =
    424. 424. Multiplication on the AL Abacus Basic facts 9  3 = 30
    425. 425. Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27
    426. 426. Multiplication on the AL Abacus Basic facts 4  8 =
    427. 427. Multiplication on the AL Abacus Basic facts 4  8 =
    428. 428. Multiplication on the AL Abacus Basic facts 4  8 =
    429. 429. Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32
    430. 430. Multiplication on the AL Abacus Basic facts 7  7 =
    431. 431. Multiplication on the AL Abacus Basic facts 7  7 =
    432. 432. Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49
    433. 433. Multiplication on the AL Abacus Commutative property 5  6 =
    434. 434. Multiplication on the AL Abacus Commutative property 5  6 =
    435. 435. Multiplication on the AL Abacus Commutative property 5  6 =
    436. 436. Multiplication on the AL Abacus Commutative property 5  6 = 6  5
    437. 437. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5
    438. 438. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5
    439. 439. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens:
    440. 440. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens:
    441. 441. Multiplication on the AL Abacus 7  8 = This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: 20 + 30
    442. 442. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: 20 + 30 50
    443. 443. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 20 + 30 50
    444. 444. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 20 + 30 50
    445. 445. Multiplication on the AL Abacus 7  8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: 3  2 20 + 30 50
    446. 446. Multiplication on the AL Abacus 7  8 = 50 + 6 This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: <ul><li>3 </li></ul><ul><li>2 </li></ul><ul><li>6 </li></ul>20 + 30 50
    447. 447. Multiplication on the AL Abacus 7  8 = 50 + 6 = 56 This method was used in the Middle Ages, rather than memorize the facts > 5  5. For facts > 5  5 Tens: Ones: <ul><li>3 </li></ul><ul><li>2 </li></ul><ul><li>6 </li></ul>20 + 30 50
    448. 448. Multiplication on the AL Abacus 9  7 = For facts > 5  5
    449. 449. Multiplication on the AL Abacus 9  7 = For facts > 5  5
    450. 450. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens:
    451. 451. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens:
    452. 452. Multiplication on the AL Abacus 9  7 = For facts > 5  5 Tens: 40 + 20
    453. 453. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: 40 + 20 60
    454. 454. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 40 + 20 60
    455. 455. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 40 + 20 60
    456. 456. Multiplication on the AL Abacus 9  7 = 60 + For facts > 5  5 Tens: Ones: 1  3 40 + 20 60
    457. 457. Multiplication on the AL Abacus 9  7 = 60 + 3 For facts > 5  5 Tens: Ones: 40 + 20 60 <ul><li>1 </li></ul><ul><li>3 </li></ul><ul><li>3 </li></ul>
    458. 458. Multiplication on the AL Abacus 9  7 = 60 + 3 = 63 For facts > 5  5 Tens: Ones: <ul><li>1 </li></ul><ul><li>3 </li></ul><ul><li>3 </li></ul>40 + 20 60
    459. 459. The Multiplication Board 6 6  4 6 x 4 on original multiplication board. 1 2 3 4 5 6 7 8 9 10
    460. 460. The Multiplication Board 6  4 Using two colors. 1 2 3 4 5 6 7 8 9 10 6
    461. 461. The Multiplication Board 7  7 7 x 7 on original multiplication board. 1 2 3 4 5 6 7 8 9 10 7
    462. 462. The Multiplication Board 7  7 Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49. 1 2 3 4 5 6 7 8 9 10 7
    463. 463. The Multiplication Board 7  7 Less clutter.
    464. 464. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.
    465. 465. Multiples Patterns Twos 2 4 6 8 10 1 2 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    466. 466. Multiples Patterns Twos 2 4 6 8 10 12 1 4 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    467. 467. Multiples Patterns Twos 2 4 6 8 10 12 14 1 6 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    468. 468. Multiples Patterns Twos 2 4 6 8 10 12 14 16 1 8 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    469. 469. Multiples Patterns Twos 2 4 6 8 1 0 12 14 16 18 2 0 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    470. 470. Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40
    471. 471. Multiples Patterns Fours 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 The ones repeat in the second row.
    472. 472. 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
    473. 473. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 16 24 32 40 48 56 64 72 80
    474. 474. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 Again the ones repeat in the second row.
    475. 475. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 4 0 48 56 64 72 8 0 The ones in the 8s show the multiples of 2.
    476. 476. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 3 2 4 0 48 56 64 7 2 8 0 The ones in the 8s show the multiples of 2.
    477. 477. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 2 4 3 2 4 0 48 56 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    478. 478. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 48 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    479. 479. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    480. 480. 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  4 6  4 is the fourth number (multiple).
    481. 481. 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  7 8  7 is the seventh number (multiple).
    482. 482. 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.
    483. 483. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>21 24 27 </li></ul><ul><li>30 </li></ul>The 3s have several patterns: Observe the ones.
    484. 484. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>21 24 27 </li></ul><ul><li>3 0 </li></ul>The 3s have several patterns: Observe the ones.
    485. 485. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>2 1 24 27 </li></ul><ul><li>3 0 </li></ul>The 3s have several patterns: Observe the ones.
    486. 486. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.
    487. 487. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.
    488. 488. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    489. 489. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    490. 490. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    491. 491. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    492. 492. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    493. 493. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    494. 494. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 30 The 3s have several patterns: The tens are the same in each row.
    495. 495. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    496. 496. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    497. 497. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    498. 498. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
    499. 499. Multiples Patterns Sevens 7 14 2 1 28 35 4 2 49 56 6 3 70 The 7s have the 1, 2, 3… pattern.
    500. 500. Multiples Patterns Sevens 7 1 4 2 1 28 3 5 4 2 49 5 6 6 3 70 The 7s have the 1, 2, 3… pattern.
    501. 501. Multiples Patterns Sevens 7 1 4 2 1 2 8 3 5 4 2 4 9 5 6 6 3 7 0 The 7s have the 1, 2, 3… pattern.
    502. 502. Multiples Patterns Sevens 7 1 4 2 1 28 35 42 49 56 63 70 Look at the tens.
    503. 503. Multiples Patterns Sevens 7 14 21 2 8 3 5 4 2 49 56 63 70 Look at the tens.
    504. 504. Multiples Patterns Sevens 7 14 21 28 35 42 4 9 5 6 6 3 70 Look at the tens.
    505. 505. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 7 0 Look at the tens.
    506. 506. Multiples Memory Aim: To help the players learn the multiples patterns. “ Multiples” are sometimes referred to as “skip counting.”
    507. 507. Multiples Memory Object of the game: To be the first player to collect all ten cards of a multiple in order. Aim: To help the players learn the multiples patterns.
    508. 508. Multiples Memory The 7s envelope contains 10 cards, each with one of the numbers listed. 7 14 21 28 35 42 49 56 63 70
    509. 509. Multiples Memory The 8s envelope contains 10 cards, each with one of the numbers listed. 8 16 24 32 40 48 56 64 72 80
    510. 510. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Players may refer to their envelopes at all times. 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    511. 511. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Players may refer to their envelopes at all times. 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    512. 512. Multiples Memory 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 Players may refer to their envelopes at all times. 7 14 21 28 35 42 49 56 63 70
    513. 513. Multiples Memory 14 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 The 7s player is looking for a 7. 7 14 21 28 35 42 49 56 63 70
    514. 514. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Wrong card, so it is turned face down in its original space. 7 14 21 28 35 42 49 56 63 70
    515. 515. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 The 8s player takes a turn. 8 16 24 32 40 48 56 64 72 80
    516. 516. Multiples Memory 40 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Cannot use this card yet. 8 16 24 32 40 48 56 64 72 80
    517. 517. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Card returned. 8 16 24 32 40 48 56 64 72 80
    518. 518. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 7 14 21 28 35 42 49 56 63 70
    519. 519. Multiples Memory 8 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 7 14 21 28 35 42 49 56 63 70
    520. 520. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 7 14 21 28 35 42 49 56 63 70
    521. 521. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    522. 522. Multiples Memory 8 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    523. 523. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 14 21 28 35 42 49 56 63 70 The needed card is collected. Receives another turn. 8 16 24 32 40 48 56 64 72 80
    524. 524. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 56 7 14 21 28 35 42 49 56 63 70 Who needs 56? [both 7s and 8s] At least one card per game is a duplicate. 8 16 24 32 40 48 56 64 72 80
    525. 525. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    526. 526. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 14 21 28 35 42 49 56 63 70 7 14 21 28 35 42 49 56 63 70
    527. 527. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 7 14 21 28 35 42 49 56 63 70 The needed card. 7 14 21 28 35 42 49 56 63 70
    528. 528. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 7 14 21 28 35 42 49 56 63 70 Where is that 14? 7 14 21 28 35 42 49 56 63 70
    529. 529. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 14 7 14 21 28 35 42 49 56 63 70 7 14 21 28 35 42 49 56 63 70
    530. 530. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 7 14 21 28 35 42 49 56 63 70 14 7 14 21 28 35 42 49 56 63 70
    531. 531. Multiples Memory 8 16 24 32 40 48 56 64 72 80 8 8 7 24 7 14 21 28 35 42 49 56 63 70 A another turn. 14 7 14 21 28 35 42 49 56 63 70
    532. 532. Multiples Memory 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 We’ll never know who won. 8 8 7 14 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80
    533. 533. Multiples Memory 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 We’ll never know who won. 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70
    534. 534. Fraction Chart 1 Giving the child the big picture, a Montessori principle. 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
    535. 535. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 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
    536. 536. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 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
    537. 537. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 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
    538. 538. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 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
    539. 539. Fraction Chart 1 How many fourths in a whole? Giving the child the big picture, a Montessori principle. 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
    540. 540. Fraction Chart 1 How many eighths in a whole? 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
    541. 541. Fraction Chart 1 Which is more, 3/4 or 4/5? 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
    542. 542. Fraction Chart 1 Which is more, 3/4 or 4/5? 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
    543. 543. Fraction Chart 1 Which is more, 3/4 or 4/5? 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
    544. 544. Fraction Chart 1 Which is more, 3/4 or 4/5? 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
    545. 545. Fraction Chart Stairs (Unit fractions) 1 1 2 1 3 1 4 1 5 1 7 1 8 1 10 1 6 1 9
    546. 546. Fraction Chart A hyperbola. Stairs (Unit fractions) 1 1 2 1 3 1 4 1 5 1 7 1 8 1 10 1 6 1 9
    547. 547. Fraction Chart 1 9/8 is 1 and 1/8. 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
    548. 548. “ Pie” Model Are we comparing angles, arcs, or area?
    549. 549. “ Pie” Model Try to compare 4/5 and 5/6 with this model. 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
    550. 550. “ Pie” Model 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
    551. 551. “ Pie” Model 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. statcan.ca
    552. 552. “ Pie” Model Difficulties
    553. 553. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul>
    554. 554. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul><ul><li>Requires counting pieces. </li></ul>
    555. 555. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul><ul><li>Requires counting pieces. </li></ul><ul><li>It does not give child the “big picture.” </li></ul>
    556. 556. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul><ul><li>Requires counting pieces. </li></ul><ul><li>It does not give child the “big picture.” </li></ul><ul><li>A fraction is much more than “a part of a set of part of a whole.” </li></ul>
    557. 557. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul><ul><li>Requires counting pieces. </li></ul><ul><li>It does not give child the “big picture.” </li></ul><ul><li>A fraction is much more than “a part of a set of part of a whole.” </li></ul><ul><li>Difficult for the child to see how fractions relate to each other. </li></ul>
    558. 558. “ Pie” Model Difficulties <ul><li>Perpetuates cultural myth fractions always < 1. </li></ul><ul><li>Requires counting pieces. </li></ul><ul><li>It does not give child the “big picture.” </li></ul><ul><li>A fraction is much more than “a part of a set of part of a whole.” </li></ul><ul><li>Difficult for the child to see how fractions relate to each other. </li></ul><ul><li>Is the user comparing angles, arcs, or area? </li></ul>
    559. 559. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8
    560. 560. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8
    561. 561. Fraction War
    562. 562. Fraction War Especially useful for learning to read a ruler with inches.
    563. 563. Fraction War 1 1 2 1 2 1 4 1 4 1 4 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 8
    564. 564. Simplifying Fractions
    565. 565. Simplifying Fractions
    566. 566. Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2.
    567. 567. Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2.
    568. 568. Simplifying Fractions In what column would you put 21/28? 21 28
    569. 569. Simplifying Fractions In what column would you put 21/28? 21 28
    570. 570. Simplifying Fractions In what column would you put 21/28? 21 28
    571. 571. Simplifying Fractions 21 28 45 72
    572. 572. Simplifying Fractions 21 28 45 72
    573. 573. Simplifying Fractions 21 28 45 72
    574. 574. Simplifying Fractions 12 16
    575. 575. Simplifying Fractions 12 16
    576. 576. Simplifying Fractions 6/8 needs further simplifying. 12 16
    577. 577. Simplifying Fractions 6/8 needs further simplifying. 12 16
    578. 578. Simplifying Fractions 6/8 needs further simplifying. 12 16
    579. 579. Simplifying Fractions 12/16 could have put here originally. 12 16
    580. 580. In Conclusion
    581. 581. In Conclusion <ul><li>We need to use quantity, not counting words, as the basis of arithmetic. </li></ul>
    582. 582. In Conclusion <ul><li>We need to use quantity, not counting words, as the basis of arithmetic. </li></ul><ul><li>Subitizing needs to be encouraged. </li></ul>
    583. 583. In Conclusion <ul><li>We need to use quantity, not counting words, as the basis of arithmetic. </li></ul><ul><li>Subitizing needs to be encouraged. </li></ul><ul><li>Children need to have visual images based on fives to remember the facts. </li></ul>
    584. 584. In Conclusion <ul><li>We need to use quantity, not counting words, as the basis of arithmetic. </li></ul><ul><li>Subitizing needs to be encouraged. </li></ul><ul><li>Children need to have visual images based on fives to remember the facts. </li></ul><ul><li>Visualizing helps our disadvantaged children because it reduces the heavy memory load. </li></ul>
    585. 585. In Conclusion <ul><li>We need to use quantity, not counting words, as the basis of arithmetic. </li></ul><ul><li>Subitizing needs to be encouraged. </li></ul><ul><li>Children need to have visual images based on fives to remember the facts. </li></ul><ul><li>Visualizing helps our disadvantaged children because it reduces the heavy memory load. </li></ul><ul><li>We need to use the math way of number naming for a longer period of time. </li></ul>
    586. 586. Adding Visualization to Montessori Mathematics IMF Conference October 21, 2011 Sarasota, Florida by Joan A. Cotter, Ph.D. [email_address] Presentation available: ALabacus.com 7 x 7 VII 1000 10 1 100 7 3 7 3

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