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NDMA Presentation

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  • 1. RightStart™ Mathematics in a
 Montessori Environment by Joan A. Cotter, Ph.D.
 JoanCotter@RightStartMath.com" 7 x 7 1000 3 2 5 5 100 10 1 New Discoveries !Montessori Academy! August 31, 2012
Hutchinson, Minnesota Other presentations available: rightstartmath.com© Joan A. Cotter, Ph.D., 2012
  • 2. National Math Crisis © Joan A. Cotter, Ph.D., 2012
  • 3. National Math Crisis•  25% of college freshmen take remedial math. © Joan A. Cotter, Ph.D., 2012
  • 4. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. © Joan A. Cotter, Ph.D., 2012
  • 5. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) © Joan A. Cotter, Ph.D., 2012
  • 6. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. © Joan A. Cotter, Ph.D., 2012
  • 7. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. •  U.S. students, compared to the world, score high at4th grade, average at 8th, and near bottom at 12th. © Joan A. Cotter, Ph.D., 2012
  • 8. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. •  U.S. students, compared to the world, score high at4th grade, average at 8th, and near bottom at 12th. •  Ready, Willing, and Unable to Serve says that 75% of17 to 24 year-olds are unfit for military service. (2010) © Joan A. Cotter, Ph.D., 2012
  • 9. Math Education is Changing © Joan A. Cotter, Ph.D., 2012
  • 10. Math Education is Changing•  The field of mathematics is doubling every 7 years. © Joan A. Cotter, Ph.D., 2012
  • 11. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. © Joan A. Cotter, Ph.D., 2012
  • 12. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. © Joan A. Cotter, Ph.D., 2012
  • 13. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. © Joan A. Cotter, Ph.D., 2012
  • 14. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. •  Calculators and computers have made computationwith many digits an unneeded skill. © Joan A. Cotter, Ph.D., 2012
  • 15. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. •  Calculators and computers have made computationwith many digits an unneeded skill. •  There is a greater emphasis on STEM subjects. © Joan A. Cotter, Ph.D., 2012
  • 16. Counting Model © Joan A. Cotter, Ph.D., 2012
  • 17. Counting Model From a childs perspectiveBecause 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 © Joan A. Cotter, Ph.D., 2012
  • 18. Counting ModelFrom a childs perspective F +E! © Joan A. Cotter, Ph.D., 2012
  • 19. Counting Model From a childs perspective F +E!A © Joan A. Cotter, Ph.D., 2012
  • 20. Counting Model From a childs perspective F +E!A B © Joan A. Cotter, Ph.D., 2012
  • 21. Counting Model From a childs perspective F +E!A B C © Joan A. Cotter, Ph.D., 2012
  • 22. Counting Model From a childs perspective F +E!A B C D E F © Joan A. Cotter, Ph.D., 2012
  • 23. Counting Model From a childs perspective F +E!A B C D E F A © Joan A. Cotter, Ph.D., 2012
  • 24. Counting Model From a childs perspective F +E!A B C D E F A B © Joan A. Cotter, Ph.D., 2012
  • 25. Counting Model From a childs perspective F +E!A B C D E F A B C D E © Joan A. Cotter, Ph.D., 2012
  • 26. Counting Model From a childs perspective F +E!A B C D E F A B C D E What is the sum?! (It must be a letter.)! © Joan A. Cotter, Ph.D., 2012
  • 27. Counting Model From a childs perspective F +E KA B C D E F G H I J K © Joan A. Cotter, Ph.D., 2012
  • 28. Counting Model From a childs perspective Now memorize the facts!! G! +D! © Joan A. Cotter, Ph.D., 2012
  • 29. Counting Model From a childs perspective Now memorize the facts!! G! +D! © Joan A. Cotter, Ph.D., 2012
  • 30. Counting Model From a childs perspective Now memorize the facts!! G! +D! D!+C! © Joan A. Cotter, Ph.D., 2012
  • 31. Counting Model From a childs perspective Now memorize the facts!! G! +D! D! C!+C! +G! © Joan A. Cotter, Ph.D., 2012
  • 32. Counting Model From a childs perspective Now memorize the facts!! G! +D! D! C!+C! +G! © Joan A. Cotter, Ph.D., 2012
  • 33. Counting Model From a childs perspective Try subtracting Hby “taking away” – E © Joan A. Cotter, Ph.D., 2012
  • 34. Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. © Joan A. Cotter, Ph.D., 2012
  • 35. Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D  E? © Joan A. Cotter, Ph.D., 2012
  • 36. Counting Model From a childs perspective L is written AB!because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  • 37. Counting Model From a childs perspective L is written AB!because it is A J !and B A’s ! huh? © Joan A. Cotter, Ph.D., 2012
  • 38. Counting Model From a childs perspective L (twelve) is written AB!because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  • 39. Counting Model From a childs perspective L (twelve) is written AB! (12) because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  • 40. Counting Model From a childs perspective L (twelve) is written AB! (12) because it is A J !(one 10) and B A’s ! © Joan A. Cotter, Ph.D., 2012
  • 41. Counting Model From a childs perspective L (twelve) is written AB! (12) because it is A J !(one 10) and B A’s(two 1s). ! © Joan A. Cotter, Ph.D., 2012
  • 42. Counting ModelIn Montessori, counting is pervasive: •  Number Rods •  Spindle Boxes •  Decimal materials •  Snake Game •  Dot Game •  Stamp Game •  Multiplication Board •  Bead Frame © Joan A. Cotter, Ph.D., 2012
  • 43. Counting Model Summary © Joan A. Cotter, Ph.D., 2012
  • 44. Counting Model Summary •  Is not natural; it takes years of practice. © Joan A. Cotter, Ph.D., 2012
  • 45. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. © Joan A. Cotter, Ph.D., 2012
  • 46. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. © Joan A. Cotter, Ph.D., 2012
  • 47. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. •  Is very error prone. © Joan A. Cotter, Ph.D., 2012
  • 48. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. •  Is very error prone. •  Is tedious and time-consuming. © Joan A. Cotter, Ph.D., 2012
  • 49. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. •  Is very error prone. •  Is tedious and time-consuming. •  Does not provide an efficient wayto master the facts. © Joan A. Cotter, Ph.D., 2012
  • 50. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 14! 15! 16! 17! 18! 19! 20! 21! 22! 23! 24! 25! 26! 27! 28! 29! 30! 31!Sometimes calendars are used for counting.! © Joan A. Cotter, Ph.D., 2012
  • 51. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 14! 15! 16! 17! 18! 19! 20! 21! 22! 23! 24! 25! 26! 27! 28! 29! 30! 31!Sometimes calendars are used for counting.! © Joan A. Cotter, Ph.D., 2012
  • 52. Calendar Math August 1! 2! 3! 4! 5! 6! 7!8! 9! 10! 11! 12! 13! 14!15! 16! 17! 18! 19! 20! 21!22! 23! 24! 25! 26! 27! 28!29! 30! 31! © Joan A. Cotter, Ph.D., 2012
  • 53. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 14! 15! 16! 17! 18! 19! 20! 21! 22! 23! 24! 25! 26! 27! 28! 29! 30! 31!This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.! 2012 © Joan A. Cotter, Ph.D.,
  • 54. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 14! 15! 16! 17! 18! 19! 20! 21! 22! 23! 24! 25! 26! 27! 28! 29! 30! 31!This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and Joan A. Cotter, Ph.D., 2012 © 3.!
  • 55. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 11! 12! 13! 14! 15! 16! 17! 18! 19! 20! 21! 22! 23! 24! 25! 26! 27! 28! 29! 30! 31! 1 2 3 4 5 6A calendar is NOT a ruler. On a ruler the numbers are not in the spaces.!Cotter, Ph.D., 2012 © Joan A.
  • 56. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10!Always show the whole calendar. A child needs to see the wholebefore the parts. Children also need to learn to plan ahead.! © Joan A. Cotter, Ph.D., 2012
  • 57. Calendar Math The calendar is not a number line. •  No quantity is involved. •  Numbers are in spaces, not at lines like a ruler. © Joan A. Cotter, Ph.D., 2012
  • 58. Calendar Math The calendar is not a number line. •  No quantity is involved. •  Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. •  Purpose of calendar is to plan ahead. •  Many ways to show the current date. © Joan A. Cotter, Ph.D., 2012
  • 59. Calendar Math The calendar is not a number line. •  No quantity is involved. •  Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. •  Purpose of calendar is to plan ahead. •  Many ways to show the current date. Calendars give a narrow view of patterning. •  Patterns do not necessarily involve numbers. •  Patterns rarely proceed row by row. •  Patterns go on forever; they don’t stop at 31. © Joan A. Cotter, Ph.D., 2012
  • 60. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 61. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 62. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 63. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 64. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 65. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  • 66. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp © Joan A. Cotter, Ph.D., 2012
  • 67. Memorizing Math 9 ! +7 ! Flash cards: © Joan A. Cotter, Ph.D., 2012
  • 68. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. © Joan A. Cotter, Ph.D., 2012
  • 69. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. © Joan A. Cotter, Ph.D., 2012
  • 70. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. •  Don’t work for those with learning disabilities. © Joan A. Cotter, Ph.D., 2012
  • 71. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. •  Don’t work for those with learning disabilities. •  Give the false impression that math isn’t aboutthinking. © Joan A. Cotter, Ph.D., 2012
  • 72. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. •  Don’t work for those with learningdisabilities. •  Give the false impression that math isn’t aboutthinking. •  Often produce stress – children under stressstop learning. © Joan A. Cotter, Ph.D., 2012
  • 73. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. •  Don’t work for those with learningdisabilities. •  Give the false impression that math isn’t aboutthinking. •  Often produce stress – children under stressstop learning. •  Are not concrete – use abstract symbols. © Joan A. Cotter, Ph.D., 2012
  • 74. Research on Counting Karen Wynn’s research Show the baby two teddy bears. ! © Joan A. Cotter, Ph.D., 2012
  • 75. Research on Counting Karen Wynn’s research Then hide them with a screen.! © Joan A. Cotter, Ph.D., 2012
  • 76. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.! © Joan A. Cotter, Ph.D., 2012
  • 77. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.! © Joan A. Cotter, Ph.D., 2012
  • 78. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.! Joan A. Cotter, Ph.D., 2012 ©
  • 79. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.! A. Cotter, Ph.D., 2012 © Joan
  • 80. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.! 2012 © Joan A. Cotter, Ph.D.,
  • 81. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
  • 82. Research on Counting Other research •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. These groups matched quantities without using counting words.! © Joan A. Cotter, Ph.D., 2012
  • 83. Research on Counting Other research •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. These groups matched quantities without using counting words.! © Joan A. Cotter, Ph.D., 2012
  • 84. Research on Counting Other research •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. These groups matched quantities without using counting words.! © Joan A. Cotter, Ph.D., 2012
  • 85. Research on Counting Other research •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. •  Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. These groups matched quantities without using counting words.! © Joan A. Cotter, Ph.D., 2012
  • 86. Research on Counting In Japanese schools: •  Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
  • 87. Research on Counting In Japanese schools: •  Children are discouraged from usingcounting for adding. •  They consistently group in 5s. © Joan A. Cotter, Ph.D., 2012
  • 88. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. © Joan A. Cotter, Ph.D., 2012
  • 89. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. © Joan A. Cotter, Ph.D., 2012
  • 90. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth © Joan A. Cotter, Ph.D., 2012
  • 91. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth •  Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit © Joan A. Cotter, Ph.D., 2012
  • 92. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth •  Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit •  Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld © Joan A. Cotter, Ph.D., 2012
  • 93. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. © Joan A. Cotter, Ph.D., 2012
  • 94. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. © Joan A. Cotter, Ph.D., 2012
  • 95. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. •  Children who use their fingers as representationaltools perform better in mathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  • 96. Visualizing Mathematics © Joan A. Cotter, Ph.D., 2012
  • 97. Visualizing Mathematics“In our concern about the memorization ofmath facts or solving problems, we must notforget that the root of mathematical study isthe creation of mental pictures in theimagination and manipulating those imagesand relationships using the power of reasonand logic.” Mindy Holte (E1) © Joan A. Cotter, Ph.D., 2012
  • 98. Visualizing Mathematics “Think in pictures, because thebrain remembers images betterthan it does anything else.”   Ben Pridmore, World Memory Champion, 2009 © Joan A. Cotter, Ph.D., 2012
  • 99. Visualizing Mathematics“Mathematics is the activity ofcreating relationships, many of whichare based in visual imagery.” Wheatley and Cobb © Joan A. Cotter, Ph.D., 2012
  • 100. Visualizing Mathematics“The process of connecting symbols toimagery is at the heart of mathematicslearning.” Dienes © Joan A. Cotter, Ph.D., 2012
  • 101. Visualizing Mathematics“The role of physical manipulativeswas to help the child form thosevisual images and thus to eliminatethe need for the physicalmanipulatives.” Ginsberg and others © Joan A. Cotter, Ph.D., 2012
  • 102. Visualizing Mathematics Japanese criteria for manipulatives •  Representative of structure of numbers. •  Easily manipulated by children. •  Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
  • 103. Visualizing Mathematics Visualizing also needed in: •  Reading •  Sports •  Creativity •  Geography •  Engineering •  Construction © Joan A. Cotter, Ph.D., 2012
  • 104. Visualizing Mathematics Visualizing also needed in: •  Reading •  Architecture •  Sports •  Astronomy •  Creativity •  Archeology •  Geography •  Chemistry •  Engineering •  Physics •  Construction •  Surgery © Joan A. Cotter, Ph.D., 2012
  • 105. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 106. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 107. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 108. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 109. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 110. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 111. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 112. Visualizing Mathematics Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 113. Visualizing Mathematics Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 114. Visualizing Mathematics Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 115. Visualizing Mathematics Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 116. Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII Romans grouped in fives. Notice 8 is 5 and 3.! © Joan A. Cotter, Ph.D., 2012
  • 117. Visualizing Mathematics : Who could read the music? Music needs 10 lines, two groups of five.! © Joan A. Cotter, Ph.D., 2012
  • 118. Research on Counting Teach Counting •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. •  Children who use their fingers as representationaltools perform better in mathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  • 119. Very Early Computation Numerals In English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eight © Joan A. Cotter, Ph.D., 2012
  • 120. Very Early Computation Numerals In English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eightIn Chinese there is only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters) © Joan A. Cotter, Ph.D., 2012
  • 121. Very Early Computation Calculating rods Because their characters are cumbersometo use for computing, the Chinese usedcalculating rods, beginning in the 4thcentury BC. © Joan A. Cotter, Ph.D., 2012
  • 122. Very Early Computation Calculating rods © Joan A. Cotter, Ph.D., 2012
  • 123. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  • 124. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  • 125. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Odd Powers of Ten) Numerals for Tens and Thousands (Odd Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  • 126. Very Early Computation Calculating rods 3578 © Joan A. Cotter, Ph.D., 2012
  • 127. Very Early Computation Calculating rods 3578 3578,3578They grouped, not in thousands, but ten-thousands! © Joan A. Cotter, Ph.D., 2012
  • 128. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 129. Naming Quantities Using fingers Naming quantities is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  • 130. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.! © Joan A. Cotter, Ph.D., 2012
  • 131. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 132. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 133. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.! © Joan A. Cotter, Ph.D., 2012
  • 134. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 135. 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.! Joan A. Cotter, Ph.D., 2012 ©
  • 136. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 137. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 138. 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.! 2012 Joan A. Cotter, Ph.D.,
  • 139. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.! © Joan A. Cotter, Ph.D., 2012
  • 140. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  • 141. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  • 142. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young childrenhave problems with diagonals.! © Joan A. Cotter, Ph.D., 2012
  • 143. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  • 144. Naming Quantities Tally sticks Start a new row for every ten.! © Joan A. Cotter, Ph.D., 2012
  • 145. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples? How would you find the answer without counting?! © Joan A. Cotter, Ph.D., 2012
  • 146. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples? 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.! © Joan A. Cotter, Ph.D., 2012
  • 147. Naming QuantitiesNumber Chart 1" 2" 3" 4" 5! © Joan A. Cotter, Ph.D., 2012
  • 148. Naming Quantities Number Chart 1" 2"To help the 3"child learnthe symbols 4" 5! © Joan A. Cotter, Ph.D., 2012
  • 149. Naming Quantities Number Chart 1" 6! 2" 7!To help the 3" 8!child learnthe symbols 4" 9! 5! 10! © Joan A. Cotter, Ph.D., 2012
  • 150. Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.! © Joan A. Cotter, Ph.D., 2012
  • 151. Naming Quantities Ordering Finger Cards Putting the finger cards in order.! © Joan A. Cotter, Ph.D., 2012
  • 152. Naming Quantities Matching Numbers to Finger Cards 5! 1! 10!Match the number to the finger card.! © Joan A. Cotter, Ph.D., 2012
  • 153. Naming Quantities Matching Fingers to Number Cards 9! 1! 10! 4! 6! 2! 3! 7! 8! 5!Match the finger card to the number.! © Joan A. Cotter, Ph.D., 2012
  • 154. Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.! © Joan A. Cotter, Ph.D., 2012
  • 155. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  • 156. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  • 157. Naming Quantities Number Rods Using different colors.! © Joan A. Cotter, Ph.D., 2012
  • 158. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate©containers.! Joan A. Cotter, Ph.D., 2012
  • 159. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.! © Joan A. Cotter, Ph.D., 2012
  • 160. Naming Quantities Spindle Box 0 1 2 3 4The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 161. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 162. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 163. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 164. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 165. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 166. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right.! © Joan A. Cotter, Ph.D., 2012
  • 167. Naming Quantities Black and White Bead Stairs “Grouped in fives so the child does not need to count.” A. M. Joosten This was the inspiration to group in 5s.! © Joan A. Cotter, Ph.D., 2012
  • 168. AL Abacus 1000 100 10 1Double-sided AL abacus. Side 1 is grouped in 5s.!Trading Side introduces algorithms with trading. ! © Joan A. Cotter, Ph.D., 2012
  • 169. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
  • 170. AL Abacus Entering quantities 3 Quantities are entered all at once, not counted.! © Joan A. Cotter, Ph.D., 2012
  • 171. AL Abacus Entering quantities 5 Relate quantities to hands.! © Joan A. Cotter, Ph.D., 2012
  • 172. AL Abacus Entering quantities 7 © Joan A. Cotter, Ph.D., 2012
  • 173. AL Abacus Entering quantities 10 © Joan A. Cotter, Ph.D., 2012
  • 174. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.!© Joan A. Cotter, Ph.D., 2012
  • 175. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  • 176. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  • 177. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  • 178. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  • 179. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  • 180. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.! © Joan A. Cotter, Ph.D., 2012
  • 181. 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.! © Joan A. Cotter, Ph.D., 2012
  • 182. 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.! © Joan A. Cotter, Ph.D., 2012
  • 183. Go to the Dump GameThe ways to partition 10.! © Joan A. Cotter, Ph.D., 2012
  • 184. Go to the Dump Game StartingA game viewed from above.! © Joan A. Cotter, Ph.D., 2012
  • 185. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 StartingEach player takes 5 cards.! © Joan A. Cotter, Ph.D., 2012
  • 186. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes YellowCap have any pairs? [no]! © Joan A. Cotter, Ph.D., 2012
  • 187. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1]! © Joan A. Cotter, Ph.D., 2012
  • 188. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1]! © Joan A. Cotter, Ph.D., 2012
  • 189. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1]! © Joan A. Cotter, Ph.D., 2012
  • 190. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2]! © Joan A. Cotter, Ph.D., 2012
  • 191. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2]! © Joan A. Cotter, Ph.D., 2012
  • 192. Go to the Dump Game 72 7 9 5 7 3 4 6 2 1 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2]! © Joan A. Cotter, Ph.D., 2012
  • 193. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2]! © Joan A. Cotter, Ph.D., 2012
  • 194. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left.! © Joan A. Cotter, Ph.D., 2012
  • 195. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left.! © Joan A. Cotter, Ph.D., 2012
  • 196. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 3 2 8 4 6 1 4 9 PlayingThe player asks the player on her left.! © Joan A. Cotter, Ph.D., 2012
  • 197. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 3? 2 7 9 5 2 8 4 6 1 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  • 198. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 PlayingYellowCap gets another turn.! © Joan A. Cotter, Ph.D., 2012
  • 199. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. PlayingYellowCap gets another turn.! © Joan A. Cotter, Ph.D., 2012
  • 200. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. Playing © Joan A. Cotter, Ph.D., 2012
  • 201. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  • 202. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Playing have a 6? © Joan A. Cotter, Ph.D., 2012
  • 203. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Go to the dump. Playing have a 6? © Joan A. Cotter, Ph.D., 2012
  • 204. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  • 205. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  • 206. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  • 207. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 1 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  • 208. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 19 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  • 209. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  • 210. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 2 9 1 7 7 5 4 9 PlayingPinkCap is not out of the game. Her turn ends, but she takes 5 moreJoan A. Cotter, Ph.D., 2012 © cards.!
  • 211. Go to the Dump Game 9 1 4 6 5 5 Winner? © Joan A. Cotter, Ph.D., 2012
  • 212. Go to the Dump Game 9 1 4 6 5 Winner?No counting. Combine both stacks.! © Joan A. Cotter, Ph.D., 2012
  • 213. Go to the Dump Game 9 1 4 6 5 Winner?Whose stack is the highest?! © Joan A. Cotter, Ph.D., 2012
  • 214. Go to the Dump Game Next gameNo shuffling needed for next game.! © Joan A. Cotter, Ph.D., 2012
  • 215. “Math” Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
  • 216. “Math” Way of Naming Numbers11 = ten 1 © Joan A. Cotter, Ph.D., 2012
  • 217. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
  • 218. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
  • 219. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 © Joan A. Cotter, Ph.D., 2012
  • 220. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  • 221. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  • 222. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  • 223. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 . . . . 19 = ten 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  • 224. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 . . . . 19 = ten 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  • 225. “Math” Way of Naming Numbers11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 . . . . . . . . 19 = ten 9 . . . . 99 = 9-ten 9 © Joan A. Cotter, Ph.D., 2012
  • 226. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way. ! © Joan A. Cotter, Ph.D., 2012
  • 227. “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. ! © Joan A. Cotter, Ph.D., 2012
  • 228. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !Shows how far children from 3 countries can count at ages 4, 5, andJoan A.!Cotter, Ph.D., 2012 © 6.
  • 229. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !Purple is Chinese. Note jump between ages 5 and 6. ! © Joan A. Cotter, Ph.D., 2012
  • 230. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !Dark green is Korean “math” way.! © Joan A. Cotter, Ph.D., 2012
  • 231. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !Dotted green is everyday Korean; notice smaller jump between ages Joan A. Cotter, 6.! 2012 © 5 and Ph.D.,
  • 232. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !Red is English speakers. They learn same amount between ages 4-5 Joan A. Cotter, Ph.D., 2012 © and 5-6.!
  • 233. Math Way of Naming Numbers •  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.) © Joan A. Cotter, Ph.D., 2012
  • 234. Math Way of Naming Numbers •  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.) •  Asian children learn mathematics using the math way of counting. © Joan A. Cotter, Ph.D., 2012
  • 235. Math Way of Naming Numbers •  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.) •  Asian children learn mathematics using the math way of counting. •  They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. © Joan A. Cotter, Ph.D., 2012
  • 236. Math Way of Naming Numbers •  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.) •  Asian children learn mathematics using the math way of counting. •  They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. •  Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. © Joan A. Cotter, Ph.D., 2012
  • 237. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
  • 238. Math Way of Naming Numbers Compared to reading: •  Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. © Joan A. Cotter, Ph.D., 2012
  • 239. Math Way of Naming Numbers Compared to reading: •  Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. •  Just as we first teach the sound of the letters, we mustfirst teach the name of the quantity (math way). © Joan A. Cotter, Ph.D., 2012
  • 240. Math Way of Naming Numbers Compared to reading: •  Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. •  Just as we first teach the sound of the letters, we mustfirst teach the name of the quantity (math way). •  Montessorians do use the math way of namingnumbers but are too quick to switch to traditionalnames. Use the math way for a longer period of time. © Joan A. Cotter, Ph.D., 2012
  • 241. Math Way of Naming Numbers“Rather, the increased gap between Chinese andU.S. students and that of Chinese Americans andCaucasian Americans may be due primarily to thenature of their initial gap prior to formal schooling,such as counting efficiency and base-ten numbersense.” Jian Wang and Emily Lin, 2005 Researchers © Joan A. Cotter, Ph.D., 2012
  • 242. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask thechild to construct 48. © Joan A. Cotter, Ph.D., 2012
  • 243. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask thechild to construct 48. Then ask the child tosubtract 14. © Joan A. Cotter, Ph.D., 2012
  • 244. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14. © Joan A. Cotter, Ph.D., 2012
  • 245. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 246. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 247. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 248. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 249. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 250. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 251. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  • 252. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  • 253. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  • 254. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  • 255. Math Way of Naming Numbers Traditional names 4-ten = fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • 256. 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.! © Joan A. Cotter, Ph.D., 2012
  • 257. Math Way of Naming Numbers Traditional names 6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • 258. Math Way of Naming Numbers Traditional names 3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  • 259. Math Way of Naming Numbers Traditional names 5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  • 260. Math Way of Naming Numbers Traditional names 2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 261. 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.! © Joan A. Cotter, Ph.D., 2012
  • 262. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news Say the syllables backward. This is how we say the teen numbers.! © Joan A. Cotter, Ph.D., 2012
  • 263. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox Say the syllables backward. This is how we say the teen numbers.! © Joan A. Cotter, Ph.D., 2012
  • 264. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 265. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 266. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 267. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  • 268. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  • 269. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  • 270. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 271. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 272. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  • 273. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  • 274. Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
  • 275. Composing Numbers 3-ten 30Point to the 3 and say 3.! © Joan A. Cotter, Ph.D., 2012
  • 276. Composing Numbers 3-ten 30Point to 0 and say 10. The 0 makes 3 a ten.! © Joan A. Cotter, Ph.D., 2012
  • 277. Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
  • 278. Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
  • 279. Composing Numbers 3-ten 7 30 7 © Joan A. Cotter, Ph.D., 2012
  • 280. Composing Numbers 3-ten 7 37 0Place the 7 on top of the 0 of the 30.! © Joan A. Cotter, Ph.D., 2012
  • 281. Composing Numbers 3-ten 7 37 0Notice the way we say the number, represent thenumber, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
  • 282. Composing Numbers 7-ten 8 78Another example.! © Joan A. Cotter, Ph.D., 2012
  • 283. Composing Numbers 10-ten © Joan A. Cotter, Ph.D., 2012
  • 284. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 285. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 286. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 287. Composing Numbers 1 hundred © Joan A. Cotter, Ph.D., 2012
  • 288. Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
  • 289. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred.! © Joan A. Cotter, Ph.D., 2012
  • 290. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred.! © Joan A. Cotter, Ph.D., 2012
  • 291. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred.! © Joan A. Cotter, Ph.D., 2012
  • 292. Composing Numbers Reading numbers backward To read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258 © Joan A. Cotter, Ph.D., 2012
  • 293. Composing Numbers Reading numbers backward To read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258 © Joan A. Cotter, Ph.D., 2012
  • 294. Composing Numbers Reading numbers backward To read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258 © Joan A. Cotter, Ph.D., 2012
  • 295. Composing Numbers Reading numbers backward To read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258 © Joan A. Cotter, Ph.D., 2012
  • 296. Composing Numbers Reading numbers backward To read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258The Decimal Cards encourage reading numbersin the normal order. © Joan A. Cotter, Ph.D., 2012
  • 297. Composing Numbers Scientific Notation 3! 4000 = 4 x 10 In scientific notation, we “stand” onthe left digit and note the number ofdigits to the right. (That’s why weshouldn’t refer to the 4 as the 4thcolumn.) © Joan A. Cotter, Ph.D., 2012
  • 298. Fact Strategies © Joan A. Cotter, Ph.D., 2012
  • 299. Fact Strategies•  A strategy is a way to learn a new fact orrecall a forgotten fact. © Joan A. Cotter, Ph.D., 2012
  • 300. Fact Strategies•  A strategy is a way to learn a new fact orrecall a forgotten fact. •  A visualizable representation is part of apowerful strategy. © Joan A. Cotter, Ph.D., 2012
  • 301. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 302. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 303. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 304. Fact Strategies Complete the Ten 9+5=Take 1 from the5 and give it tothe 9. © Joan A. Cotter, Ph.D., 2012
  • 305. 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.! © Joan A. Cotter, Ph.D., 2012
  • 306. Fact Strategies Complete the Ten 9+5=Take 1 from the5 and give it tothe 9. © Joan A. Cotter, Ph.D., 2012
  • 307. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 from the5 and give it tothe 9. © Joan A. Cotter, Ph.D., 2012
  • 308. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 309. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 310. Fact Strategies Two Fives 8+6=Two fives make 10. ! © Joan A. Cotter, Ph.D., 2012
  • 311. Fact Strategies Two Fives 8+6=Just add the “leftovers.”! © Joan A. Cotter, Ph.D., 2012
  • 312. Fact Strategies Two Fives 8+6= 10 + 4 = 14Just add the “leftovers.”! © Joan A. Cotter, Ph.D., 2012
  • 313. Fact Strategies Two Fives 7+5=Another example.! © Joan A. Cotter, Ph.D., 2012
  • 314. Fact Strategies Two Fives 7+5= © Joan A. Cotter, Ph.D., 2012
  • 315. Fact Strategies Two Fives 7 + 5 = 12 © Joan A. Cotter, Ph.D., 2012
  • 316. Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 317. Fact Strategies Difference 7–4=Subtract 4 from5; then add 2. © Joan A. Cotter, Ph.D., 2012
  • 318. Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 319. Fact Strategies Going Down 15 – 9 =Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 320. Fact Strategies Going Down 15 – 9 =Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 321. Fact Strategies Going Down 15 – 9 =Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 322. Fact Strategies Going Down 15 – 9 = 6Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 323. Fact Strategies Subtract from 10 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 324. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 325. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 326. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 327. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 328. Fact Strategies Going Up 13 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 329. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 330. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 331. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 332. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 333. Fact Strategies Going Up 13 – 9 = 1+3=4Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 334. MoneyPenny © Joan A. Cotter, Ph.D., 2012
  • 335. MoneyNickel © Joan A. Cotter, Ph.D., 2012
  • 336. Money Dime © Joan A. Cotter, Ph.D., 2012
  • 337. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 338. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 339. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 340. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 341. Base-10 Picture Cards One © Joan A. Cotter, Ph.D., 2012
  • 342. Base-10 Picture Cards Ten One © Joan A. Cotter, Ph.D., 2012
  • 343. Base-10 Picture Cards Hundred Ten One © Joan A. Cotter, Ph.D., 2012
  • 344. Base-10 Picture Cards Thousand Hundred Ten One © Joan A. Cotter, Ph.D., 2012
  • 345. Base-10 Picture Cards Add using the base-10 picture cards. 3658! +2724! © Joan A. Cotter, Ph.D., 2012
  • 346. Base-10 Picture Cards 6 5 8 3 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 347. Base-10 Picture Cards 6 5 8 3 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 348. Base-10 Picture Cards 6 5 8 3 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 349. Base-10 Picture Cards 6 5 8 3 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 350. Base-10 Picture Cards 7 2 4 2 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 351. Base-10 Picture Cards 7 2 4 2 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 352. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Add them together. © Joan A. Cotter, Ph.D., 2012
  • 353. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 354. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 355. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 356. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 357. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 358. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0Trade 10 hundreds for 1 thousand. © Joan A. Cotter, Ph.D., 2012
  • 359. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0Trade 10 hundreds for 1 thousand. © Joan A. Cotter, Ph.D., 2012
  • 360. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0Trade 10 hundreds for 1 thousand. © Joan A. Cotter, Ph.D., 2012
  • 361. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0Trade 10 hundreds for 1 thousand. © Joan A. Cotter, Ph.D., 2012
  • 362. Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 3 8 2 6 0 0 0 © Joan A. Cotter, Ph.D., 2012
  • 363. Bead Frame 1 10 1001000 © Joan A. Cotter, Ph.D., 2012
  • 364. Bead Frame 1 8 10 + 6 100 1000 © Joan A. Cotter, Ph.D., 2012
  • 365. Bead Frame 1 8 10 + 6 100 1000 © Joan A. Cotter, Ph.D., 2012
  • 366. Bead Frame 1 8 10 + 6 100 1000 © Joan A. Cotter, Ph.D., 2012
  • 367. Bead Frame 1 8 10 + 6 100 1000 © Joan A. Cotter, Ph.D., 2012
  • 368. Bead Frame 1 8 10 + 6 100 1000 © Joan A. Cotter, Ph.D., 2012
  • 369. Bead Frame 1 8 10 + 6 1001000 © Joan A. Cotter, Ph.D., 2012
  • 370. Bead Frame 1 8 10 + 6 1001000 © Joan A. Cotter, Ph.D., 2012
  • 371. Bead Frame 1 8 10 + 6 1001000 © Joan A. Cotter, Ph.D., 2012
  • 372. Bead Frame 1 8 10 + 6 1001000 © Joan A. Cotter, Ph.D., 2012
  • 373. Bead Frame 1 8 10 + 6 100 14 1000 © Joan A. Cotter, Ph.D., 2012
  • 374. 1 Bead Frame 10 100 1000Difficulties for the child © Joan A. Cotter, Ph.D., 2012
  • 375. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. © Joan A. Cotter, Ph.D., 2012
  • 376. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. •  Not visualizable: Beads need to be grouped in fives. © Joan A. Cotter, Ph.D., 2012
  • 377. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. •  Not visualizable: Beads need to be grouped in fives. •  When beads are moved right, inconsistent withequation order: Beads need to be moved left. © Joan A. Cotter, Ph.D., 2012
  • 378. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. •  Not visualizable: Beads need to be grouped in fives. •  When beads are moved right, inconsistent withequation order: Beads need to be moved left. •  Hierarchies of numbers represented sideways:They need to be in vertical columns. © Joan A. Cotter, Ph.D., 2012
  • 379. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. •  Not visualizable: Beads need to be grouped in fives. •  When beads are moved right, inconsistent withequation order: Beads need to be moved left. •  Hierarchies of numbers represented sideways:They need to be in vertical columns. •  Trading done before second number is completelyadded: Addends need to be combined before trading. © Joan A. Cotter, Ph.D., 2012
  • 380. 1 Bead Frame 10 100 1000 Difficulties for the child •  Distracting: Room is visible through the frame. •  Not visualizable: Beads need to be grouped in fives. •  When beads are moved right, inconsistent withequation order: Beads need to be moved left. •  Hierarchies of numbers represented sideways:They need to be in vertical columns. •  Trading done before second number is completelyadded: Addends need to be combined before trading. •  Answer is read going up: We read top to bottom. © Joan A. Cotter, Ph.D., 2012
  • 381. Trading Side Cleared 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 382. Trading Side Thousands 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 383. Trading Side Hundreds 1000 100 10 1The third wire from each end is not used.! © Joan A. Cotter, Ph.D., 2012
  • 384. Trading Side Tens 1000 100 10 1The third wire from each end is not used.! © Joan A. Cotter, Ph.D., 2012
  • 385. Trading Side Ones 1000 100 10 1The third wire from each end is not used.! © Joan A. Cotter, Ph.D., 2012
  • 386. Trading Side Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 387. Trading Side Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 388. Trading Side Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 389. Trading Side Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 390. Trading Side Adding 1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  • 391. Trading Side Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow).! © Joan A. Cotter, Ph.D., 2012
  • 392. Trading Side Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 393. Trading Side Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 394. Trading Side Adding 1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
  • 395. Trading Side Cleared 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 396. Trading Side Bead Trading game 1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 397. Trading Side Bead Trading game 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 398. Trading Side Bead Trading game 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 399. Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade?! © Joan A. Cotter, Ph.D., 2012
  • 400. Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade?! © Joan A. Cotter, Ph.D., 2012
  • 401. Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade?! © Joan A. Cotter, Ph.D., 2012
  • 402. Trading Side Bead Trading game 1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 403. Trading Side Bead Trading game 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 404. Trading Side Bead Trading game 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 405. Trading Side Bead Trading game 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 406. Trading Side Bead Trading game 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 407. Trading Side Bead Trading game 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 408. Trading Side Bead Trading game 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 409. Trading Side Bead Trading game 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 410. Trading Side Bead Trading game 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 411. Trading Side Bead Trading game •  In the Bead Trading game 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
  • 412. Trading Side Bead Trading game •  In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; © Joan A. Cotter, Ph.D., 2012
  • 413. Trading Side Bead Trading game •  In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. © Joan A. Cotter, Ph.D., 2012
  • 414. Trading Side Bead Trading game •  In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience thegreater value of each column from left to right. © Joan A. Cotter, Ph.D., 2012
  • 415. Trading Side Bead Trading game •  In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience thegreater value of each column from left to right. •  To detect a pattern, there must be at least threeexamples in the sequence. (Place value is a pattern.) © Joan A. Cotter, Ph.D., 2012
  • 416. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  • 417. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 418. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 419. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 420. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 421. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 422. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 423. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 424. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 425. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 426. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 427. Trading Side Adding 4-digit numbers 1000 100 10 1 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen?! © Joan A. Cotter, Ph.D., 2012
  • 428. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. Is it okay to show the extra ten by writing a 1 above the tens column?! Cotter, Ph.D., 2012 © Joan A.
  • 429. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 430. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. Do we need to trade? [no]! © Joan A. Cotter, Ph.D., 2012
  • 431. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 432. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 433. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. Do we need to trade? [yes]! © Joan A. Cotter, Ph.D., 2012
  • 434. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. Notice the number of yellow beads. [3] Notice the number ofblue beads left. [3] Coincidence? No, because 13 – 10 = 3.! © Joan A. Cotter, Ph.D., 2012
  • 435. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 436. Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 437. Trading Side Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 438. Trading Side Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 439. Trading Side Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 440. Trading Side Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 441. Trading Side Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 442. Role of the AL Abacus “Neither is it strange to us, looking back,that there should have come a result quiteunforeseen by the educators of that time,namely, a loss of the power of real insightinto number [by not using abacuses].” David Eugene Smith, 1903 © Joan A. Cotter, Ph.D., 2012
  • 443. Role of the AL Abacus Its functions •  Provides a visual organization of quantity. •  Allows child to handle quantities in 5s and 10s. •  Along with the math way of number naming,makes place value transparent. •  Shows strategies concretely for learning facts. •  Models trading tens needed for algorithms onthe trading side. •  Physical abacus leads to developing mental abacus. © Joan A. Cotter, Ph.D., 2012
  • 444. Role of the AL Abacus Stages1. The abacus is needed for all number activities. 2. It’s used selectively for new concepts or unsure facts. 3. Beads are moved on an imaginery abacus. 4. The abacus becomes completely internalized. Since this occurs at different times for different children, they must be encouraged to use the abacus whenever they need it. © Joan A. Cotter, Ph.D., 2012
  • 445. Mental Addition You are sitting at your desk with a calculator, paper and pencil, and a box of teddy bears. You need to find twenty-four plus thirty-eight. How do you do it? Research shows a majority of people do it mentally.“How would you do it mentally?” Discuss methods.! © Joan A. Cotter, Ph.D., 2012
  • 446. Mental Addition Dutch method 24 + 38 = A very efficient way, taught to Dutch children, especially oral.! © Joan A. Cotter, Ph.D., 2012
  • 447. Mental Addition Dutch method 24 + 38 = 24 + A very efficient way, taught to Dutch children, especially oral.! © Joan A. Cotter, Ph.D., 2012
  • 448. Mental Addition Dutch method 24 + 38 = 24 + 30 + A very efficient way, taught to Dutch children, especially oral.! © Joan A. Cotter, Ph.D., 2012
  • 449. Mental Addition Dutch method 24 + 38 = 24 + 30 + 8 = A very efficient way, taught to Dutch children, especially oral.! © Joan A. Cotter, Ph.D., 2012
  • 450. Multiplication on the AL Abacus Basic facts 64= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 451. Multiplication on the AL Abacus Basic facts 64= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 452. Multiplication on the AL Abacus Basic facts 64= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 453. Multiplication on the AL Abacus Basic facts 64= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 454. Multiplication on the AL Abacus Basic facts 64= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 455. Multiplication on the AL Abacus Basic facts 93= © Joan A. Cotter, Ph.D., 2012
  • 456. Multiplication on the AL Abacus Basic facts 93= © Joan A. Cotter, Ph.D., 2012
  • 457. Multiplication on the AL Abacus Basic facts 93= 30 © Joan A. Cotter, Ph.D., 2012
  • 458. Multiplication on the AL Abacus Basic facts 93= 30 – 3 = 27 © Joan A. Cotter, Ph.D., 2012
  • 459. Multiplication on the AL Abacus Basic facts 48= © Joan A. Cotter, Ph.D., 2012
  • 460. Multiplication on the AL Abacus Basic facts 48= © Joan A. Cotter, Ph.D., 2012
  • 461. Multiplication on the AL Abacus Basic facts 48= © Joan A. Cotter, Ph.D., 2012
  • 462. Multiplication on the AL Abacus Basic facts 48= 20 + 12 = 32 © Joan A. Cotter, Ph.D., 2012
  • 463. Multiplication on the AL Abacus Basic facts 77= © Joan A. Cotter, Ph.D., 2012
  • 464. Multiplication on the AL Abacus Basic facts 77= © Joan A. Cotter, Ph.D., 2012
  • 465. Multiplication on the AL Abacus Basic facts 77= 25 + 10 + 10 + 4 = 49 © Joan A. Cotter, Ph.D., 2012
  • 466. Multiplication on the AL Abacus Commutative property 56= © Joan A. Cotter, Ph.D., 2012
  • 467. Multiplication on the AL Abacus Commutative property 56= © Joan A. Cotter, Ph.D., 2012
  • 468. Multiplication on the AL Abacus Commutative property 56= © Joan A. Cotter, Ph.D., 2012
  • 469. Multiplication on the AL Abacus Commutative property 56=65 © Joan A. Cotter, Ph.D., 2012
  • 470. The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 64 6!6 x 4 on original multiplication board.! © Joan A. Cotter, Ph.D., 2012
  • 471. The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 6  4 6!Using two colors.! © Joan A. Cotter, Ph.D., 2012
  • 472. The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 77 7!7 x 7 on original multiplication board.! © Joan A. Cotter, Ph.D., 2012
  • 473. The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 77 7!Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49.! A. Cotter, Ph.D., 2012 © Joan
  • 474. The Multiplication Board 77Less clutter.! © Joan A. Cotter, Ph.D., 2012
  • 475. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 476. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 477. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 478. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 479. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 480. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012
  • 481. Multiples Patterns Fours 4 8 12 16 2024 28 32 36 40 © Joan A. Cotter, Ph.D., 2012
  • 482. Multiples Patterns Fours 4 8 12 16 2024 28 32 36 40The ones repeat in the second row. © Joan A. Cotter, Ph.D., 2012
  • 483. Multiples Patterns Sixes and Eights 6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 © Joan A. Cotter, Ph.D., 2012
  • 484. Multiples Patterns Sixes and Eights 6 12 18 24 3036 42 48 54 608 16 24 32 4048 56 64 72 80 © Joan A. Cotter, Ph.D., 2012
  • 485. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80Again the ones repeat in the second row. © Joan A. Cotter, Ph.D., 2012
  • 486. 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
  • 487. 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
  • 488. 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
  • 489. 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
  • 490. 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
  • 491. Multiples Patterns Sixes and Eights 6 12 18 24 30 64 36 42 48 54 60 8 16 24 32 40 48 56 64 72 806  4 is the fourth number (multiple). © Joan A. Cotter, Ph.D., 2012
  • 492. 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78  7 is the seventh number (multiple). © Joan A. Cotter, Ph.D., 2012
  • 493. Multiples Patterns Nines 9 18 27 36 45 90 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
  • 494. 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
  • 495. 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
  • 496. 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
  • 497. 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
  • 498. 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
  • 499. 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
  • 500. 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
  • 501. 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
  • 502. 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
  • 503. 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
  • 504. 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
  • 505. 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
  • 506. 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
  • 507. 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
  • 508. 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
  • 509. 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
  • 510. 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
  • 511. 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
  • 512. 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
  • 513. 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
  • 514. 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
  • 515. 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
  • 516. 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
  • 517. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
  • 518. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
  • 519. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens. © Joan A. Cotter, Ph.D., 2012
  • 520. Multiples Memory“Multiples” are sometimes referred to as “skip counting.”! © Joan A. Cotter, Ph.D., 2012
  • 521. Multiples Memory Aim: To help the players learn the multiples patterns. “Multiples” are sometimes referred to as “skip counting.”! © Joan A. Cotter, Ph.D., 2012
  • 522. Multiples Memory Aim: 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
  • 523. 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
  • 524. 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
  • 525. 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 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • 526. 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 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • 527. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • 528. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 14 70The 7s player is looking for a 7. © Joan A. Cotter, Ph.D., 2012
  • 529. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Wrong card, so it is turned face down in its original space.! © Joan A. Cotter, Ph.D., 2012
  • 530. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70The 8s player takes a turn. © Joan A. Cotter, Ph.D., 2012
  • 531. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 40Cannot use this card yet. © Joan A. Cotter, Ph.D., 2012
  • 532. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Card returned. © Joan A. Cotter, Ph.D., 2012
  • 533. 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
  • 534. 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
  • 535. 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
  • 536. 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
  • 537. 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
  • 538. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8The needed card is collected. Receives another turn. © Joan A. Cotter, Ph.D., 2012
  • 539. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 56 70 8Who needs 56? [both 7s and 8s] At least one card per game is a duplicate.Ph.D., 2012 © Joan A. Cotter,
  • 540. 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
  • 541. 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
  • 542. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8The needed card. © Joan A. Cotter, Ph.D., 2012
  • 543. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7Where is that 14? © Joan A. Cotter, Ph.D., 2012
  • 544. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 14 70 87 © Joan A. Cotter, Ph.D., 2012
  • 545. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 14 © Joan A. Cotter, Ph.D., 2012
  • 546. Multiples Memory 7 14 21 28 35 42 49 56 63 70 24 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7 14A another turn. © Joan A. Cotter, Ph.D., 2012
  • 547. 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 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7 14We’ll never know who won. © Joan A. Cotter, Ph.D., 2012
  • 548. 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 40 28 35 42 48 56 64 72 80 49 56 63 70We’ll never know who won. © Joan A. Cotter, Ph.D., 2012
  • 549. 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 10 Giving the student the big picture. © Joan A. Cotter, Ph.D., 2012
  • 550. 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 10 How many fourths in a whole? Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • 551. 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 10 How many fourths in a whole? Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • 552. 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 10 How many fourths in a whole? Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • 553. 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 10 How many fourths in a whole? Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • 554. 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 10 How many fourths in a whole? Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • 555. 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 10 How many eighths in a whole? Giving the student the big picture. © Joan A. Cotter, Ph.D., 2012
  • 556. 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 10 Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
  • 557. 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 10 Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
  • 558. 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 10 Which is more, 3/4 or 4/5? Giving the child the big picture. © Joan A. Cotter, Ph.D., 2012
  • 559. 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 10 Which is more, 3/4 or 4/5? Giving the child the big picture. © Joan A. Cotter, Ph.D., 2012
  • 560. Fraction Chart!1 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
  • 561. Fraction Chart!1 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
  • 562. 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 1 8 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 10 9/8 is 1 and 1/8. ! © Joan A. Cotter, Ph.D., 2012
  • 563. Circle ModelAre we comparing angles, arcs, or area? © Joan A. Cotter, Ph.D., 2012
  • 564. Circle Model 1 1 3 3 1 1 2 2 1 3 1 1 1 1 1 1 5 5 6 6 4 4 1 1 1 1 6 6 1 1 5 5 4 4 1 1 1 6 6 5 Try to compare 4/5 and 5/6 with this model. © Joan A. Cotter, Ph.D., 2012
  • 565. Circle ModelExperts in visual literacy say that comparingquantities in pie charts is difficult becausemost people think linearly. It is easier tocompare along a straight line than comparepie slices. askoxford.com © Joan A. Cotter, Ph.D., 2012
  • 566. Circle ModelExperts in visual literacy say that comparingquantities in pie charts is difficult becausemost people think linearly. It is easier tocompare along a straight line than comparepie slices. askoxford.com Specialists also suggest refraining from usingmore than one pie chart for comparison. www.statcan.ca © Joan A. Cotter, Ph.D., 2012
  • 567. Circle Model Difficulties © Joan A. Cotter, Ph.D., 2012
  • 568. Circle Model Difficulties •  Perpetuates cultural myth fractions are 1. © Joan A. Cotter, Ph.D., 2012
  • 569. Circle Model Difficulties •  Perpetuates cultural myth fractions are 1. •  Does not give the child the “big picture.” © Joan A. Cotter, Ph.D., 2012
  • 570. Circle Model Difficulties •  Perpetuates cultural myth fractions are 1. •  Does not give the child the “big picture.” •  Limits understanding of fractions: they are more than “a part of a whole or part of a set.” © Joan A. Cotter, Ph.D., 2012
  • 571. Circle Model Difficulties •  Perpetuates cultural myth fractions are 1. •  Does not give the child the “big picture.” •  Limits understanding of fractions: they are more than “a part of a whole or part of a set.” •  Makes it difficult for the child to see how fractions relate to each other. © Joan A. Cotter, Ph.D., 2012
  • 572. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 573. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 574. Fraction War! © Joan A. Cotter, Ph.D., 2012
  • 575. Fraction War! 1 2 3 4 5 6Especially useful for learning to read a ruler with inches. © Joan A. Cotter, Ph.D., 2012
  • 576. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 577. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 18 © Joan A. Cotter, Ph.D., 2012
  • 578. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 18 © Joan A. Cotter, Ph.D., 2012
  • 579. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 18 4 © Joan A. Cotter, Ph.D., 2012
  • 580. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 18 4 © Joan A. Cotter, Ph.D., 2012
  • 581. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 18 4 Which is more, 1/8 or 1/4?! © Joan A. Cotter, Ph.D., 2012
  • 582. Fraction War 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 Which is more, 1/8 or 1/4?! © Joan A. Cotter, Ph.D., 2012
  • 583. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 58 © Joan A. Cotter, Ph.D., 2012
  • 584. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 58 © Joan A. Cotter, Ph.D., 2012
  • 585. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 5 38 4 © Joan A. Cotter, Ph.D., 2012
  • 586. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 5 38 4 © Joan A. Cotter, Ph.D., 2012
  • 587. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 5 38 4 Which is more, 5/8 or 3/4?! © Joan A. Cotter, Ph.D., 2012
  • 588. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 589. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 34 © Joan A. Cotter, Ph.D., 2012
  • 590. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 3 34 4 © Joan A. Cotter, Ph.D., 2012
  • 591. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 3 34 4 © Joan A. Cotter, Ph.D., 2012
  • 592. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 3 34 4 3 8 © Joan A. Cotter, Ph.D., 2012
  • 593. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 3 34 4 3 1 8 Which is more, 4 5/8 or 3/4?! © Joan A. Cotter, Ph.D., 2012
  • 594. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 595. Fraction War! 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 © Joan A. Cotter, Ph.D., 2012
  • 596. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 597. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 598. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100The fraction 4/8 can be reduced on the multiplication table as 1/2.© Joan A. Cotter, Ph.D., 2012
  • 599. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100The fraction 4/8 can be reduced on the multiplication table as 1/2.© Joan A. Cotter, Ph.D., 2012
  • 600. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • 601. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • 602. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • 603. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 604. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 605. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 606. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 607. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • 608. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • 609. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • 610. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • 611. Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 10012/16 could have put here originally. © Joan A. Cotter, Ph.D., 2012
  • 612. RightStart™ Mathematics in a
 Montessori Environment by Joan A. Cotter, Ph.D.
 JoanCotter@RightStartMath.com 7 x 7 1000 3 2 5 5 100 10 1 New Discoveries !Montessori Academy! August 31, 2012
Hutchinson, Minnesota Other presentations available: rightstartmath.com© Joan A. Cotter, Ph.D., 2012

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