1. Updating Montessori Fractions October 24, 2009 2:00 - 3:30 p.m. Session 6 Garden Grove, CA by Joan A. Cotter, Ph.D. [email_address] Slides/handouts: ALabacus.com

2. Fraction Chart At first ask child to remove and replace only first few rows. A linear model gives an overview and shows relationships.

3. Fraction Stairs Notice what happens as the fractions get smaller and smaller. The curve is a hyperbola.

4. Fraction Chart How many fourths in a whole? How many sixths? We use ordinal numbers, except for one-half, to name fractions.

5. Fraction Chart What is more, 1/4 or 1/3? What is more, 1/9 or 1/10?

6. Fraction Chart What is more, 1/4 or 1/3? What is more, 1/9 or 1/10?

7. Fraction Chart Which is more, 3/4 or 4/5?

8. Fraction Chart Which is more, 3/4 or 4/5? Which is more, 7/8 or 8/9?

9. Fraction Chart The pattern of 1/2, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10.

10. Fraction Chart How many fourths equal a half? Eighths? Sevenths?

11. Fraction Chart How many fourths equal a half? Eighths? Sevenths?

12. Fraction Chart What is half of a half?

13. “ Fish Tank” Model 2 5 With this model, could you compare 2/5 and 1/4? Also, children will think fractions are two numbers, but they are one number just as 37 is one number.

14. “ Words” Model This is fourths . This is thirds . The 1993 textbook using this model does not say the figures represent one. So fourths look like four. How do you compare fourths and thirds?

15. “ Pie” Model Try to compare 4/5 and 5/6 with this model.

16. “ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com Specialists also suggest refraining from using more than one pie chart for comparison. www.statcan.ca Even adults have difficulty with pie charts.

17. “ Rounded Corner” Model The middle fractions are greater than the fractions at the ends! 1 3 1 3 1 3 1 4 1 4 1 4 1 4

18. “ Color” Model 1 2 1 1 2 1 3 1 3 1 3 6 6 6 6 6 6 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 9 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 5 1 5 1 5 1 5 1 5 1 1 1 1 1 1 1 4 Notice how your eye tends to stay in the same row. Try, for example, to see how many eighths are in one half.

19. Fraction Chart

20. Definition of a Fraction A fraction is part of a set or part of a whole. Textbook definition Another common meaning of fraction is fragment or a small part. 3 2 What about ?

21. Definition of a Fraction An expression that indicates the quotient of two quantities. American Heritage Dictionary Do not use “over” as in 3 over 2. This definition is not appropriate for students, but it does emphasize the division aspect of fractions. “Over” does not indicate division.

22. Definition of a Fraction or 3 ÷ 2. 1 1 1 1 1 1 2 1 2 1 2 1 2 3 2 means three s 1 2

23. “ Missing 7ths & 9ths” Model Some curricula omit the sevenths and the ninths and add the twelfths. Look down the center. Fractions > 1/5 are always even numbers. 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 10 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 12 1 8

24. “ Missing 7ths & 9ths” Model With the sevenths and the ninths omitted, the pattern is obscured. Math is the science of patterns.

25. Concentrating on One Game A game to learn, for example, that 4 fourths and 8 eighths make a whole.

26. Concentrating on One Game 5 3 When playing a memory game, the player must say what she is looking for before turning over the second card.

27. Concentrating on One Game 3 5 2 5 Players use the fraction chart to find what they need. Don’t teach a rule.

28. Concentrating on One Game

29. Concentrating on One Game What is needed with 3/8 to make 1? [5/8] 3 8

30. Ruler Chart Especially useful for learning to read a ruler with inches.

31. Ruler Chart Horizontal lines removed.

32. Ruler Chart Fractions symbols removed.

33. Ruler Chart Students are often surprised to see how a ruler is constructed.

34. Ruler War Game A comparison game, using cards with ones, halves, fourths, and eighths.

35. Ruler War Game Which is more, 1/8 or 1/4? 1 4 1 8

36. Ruler War Game 3 4 5 8

37. Ruler War Game 3 4 3 4 3 8 1 4

38. Ruler War Game 3 4 3 4 3 8 1 4

39. Fraction Chart Showing 9/8 is 1 plus 1/8.

40. Mixed to Improper Fractions Each row of connected rectangles represents 1. Ask the student how she found the 11. 2 4 two 4s 3 + 3 11 = 11

41. Mixed to Improper Fractions 2 4 two 4s 11 3 four 3s + 2 = 14 Each row of connected rectangles represents 1. + 3 = 11

42. Mixed to Improper Fractions Each row of connected rectangles represents 1. 2 4 two 4s 11 3 four 5s + 3 = 23 four 3s + 2 = 14 + 3 = 11

43. Improper to Mixed Fractions

44. Improper to Mixed The correlation to division becomes obvious here.

45. Improper to Mixed The correlation to division becomes obvious here.

46. Fraction of Geometric Figures 1 2 2 3 1 4

47. Fraction of Geometric Figures 1 2 2 3 1 4

48. Fraction of Geometric Figures 1 2 2 3 1 4

49. Fraction of Geometric Figures 1 2 2 3 1 4 A study showed that many students and adults though this was impossible.

50. Making the Whole

51. Making the Whole

52. Making the Whole

53. Fraction Chart What is one-half of 12? What is one-fourth of 12?

54. Percents Percent means per hundred or out of 100.

55. Percents Percent means per hundred or out of 100. 2 of 100 = = 50% 100 1 50

56. Percents Percent means per hundred or out of 100. 4 of 100 = = 25% 100 1 25

57. Dividing 100 100 50 50 25 25 25 25 10 10 10 10 10 10 10 10 10 10 33 1 3 1 3 33 1 3 1 3 33 1 3 1 3 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2 12 1 2 1 2

58. Simplifying Fractions 1 2

59. Simplifying Fractions 3 6 = 1 2

60. Simplifying Fractions 4 8 = 1 2

61. Simplifying Fractions 8 12 = 2 3 4 Writing the common multiple in a circle, for example, the 4, helps students remember what they’re dividing by.

62. Simplifying Fractions 9 12 = 3 4 3 Writing the common multiple in a circle, in this example, 3, helps students remember what they’re dividing by.

63. Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2. 21 28 45 72

64. Simplifying Fractions 12 16

65. Skip Counting Patterns Twos Recognizing multiples necessary for simplifying fractions and doing algebra. 2 2 4 4 6 6 8 8 0 0

66. Skip Counting Patterns Fours Notice the ones repeat in the second row. 4 4 8 8 2 2 6 6 0 0

67. Skip Counting Patterns Sixes and Eights Second row repeats with the 6s and 8s. Also, the ones in the eights are counting by 2s backward, 8, 6, 4, 2, 0. 6 6 2 2 8 8 4 4 0 0 8 8 6 6 4 4 2 2 0 0

68. © Joan A. Cotter, 2009 Skip Counting Patterns Sixes and Eights 6x4 8x7 6 x 4 is the fourth number (multiple).

69. Skip Counting Patterns Nines Second row done backwards to see digits reversing. Also the digits in each number add to 9. 9 18 27 36 45 90 81 72 63 54

70. Skip Counting Patterns 15 5 12 18 21 24 27 3 6 9 30 Threes 2 8 1 4 7 3 6 9 0 Threes have several patterns. First see 0, 1, 2, 3, . . . 9.

71. Skip Counting Patterns 1 2 1 5 1 8 2 1 2 4 2 7 3 6 9 30 Threes The tens in each column are 0, 1, 2.

72. Skip Counting Patterns 6 15 24 6 12 21 3 30 Threes The second row. [6] And the third row–the nines. Now add the digits in each number in the first row. [3] 18 27 9 18 27 9 12 21 3 30 15 24 6

73. Skip Counting Patterns Sevens 28 35 42 49 56 63 7 14 21 70 Start in the upper right to see the 1, 2, 3 pattern. 8 9 7 0 5 6 4 2 3 1

74. Skip Counting Memory Game The envelope contains 10 cards, each with one of the numbers listed. 7 14 21 28 35 42 49 56 63 70 A game for learning the multiples.

75. Skip Counting Memory Game 7 14 21 28 35 42 49 56 63 70 Remove the cards from each envelope, shuffle slightly, and lay out face down. 8 16 24 32 40 48 56 64 72 80

76. Skip Counting Memory Game 14 40 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70 Players must collect the sets in order . Only one card is turned over per turn.

77. Skip Counting Memory Game 8 8 8 16 24 32 40 48 56 64 72 80 7 14 21 28 35 42 49 56 63 70

78. Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 56 7 7 14 21 28 35 42 49 56 63 70 Who needs 56? [both 7s and 8s] At least one card per game is a duplicate.

79. Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 7 14 7 14 21 28 35 42 49 56 63 70

80. Skip Counting Memory Game 8 16 24 32 40 48 56 64 72 80 8 8 7 24 21 7 14 21 28 35 42 49 56 63 70 14

81. Skip Counting Memory Game 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 We’ll never know who won. 8 8 7 14 24 21

82. © Joan A. Cotter, 2009 Subtracting Fractions 4684 – 2372 4684 – 2879 2000 300 10 2 2312 2000 – 200 10 – 5 1805 4 thousand minus 2 thousand is 2 thousand. . . .

83. © Joan A. Cotter, 2009 Subtracting Fractions 3 5 4 – 3 2 Using the previous subtraction method. 3 2 5 – 4 7 3 7 5 7 5 – 2 1 7

84. © Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 The square represents 1.

85. © Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 We are thinking 1/2 of 1/2. First find 1/2 of it vertically.

86. © Joan A. Cotter, 2009 Multiplying Fractions 1 2 x = 1 2 1 4 Now find 1/2 of it horizontally. The solution is the double crosshatched area.

87. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 Another example.

88. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4

89. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4

90. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = = 3 4 6 12 1 2

91. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 The total number of of rectangles is 3 x 4.

92. © Joan A. Cotter, 2009 Multiplying Fractions 2 3 x = 3 4 The number of double crosshatched rectangles is 2 x 3. The total number of rectangles is 3 x 4. That’s why to multiply fractions, we multiply the numerators and the denominators.

93. Dividing Fractions One meaning is how many 2s in 6. 6 ÷ 2 = __ Relating to division.

94. Dividing Fractions One meaning is how many 2s in 6. 6 ÷ 2 = __

95. Dividing Fractions Since 12 is twice as much as 6, 12 ÷ 2 is twice as much as 6 ÷ 2. 12 ÷ 2 = __

96. Dividing Fractions So 12 ÷ 2 = 2 x (6 ÷ 2) = 2 x 3 = 6. 12 ÷ 2 = __ Since 12 is twice as much as 6, 12 ÷ 2 is twice as much as 6 ÷ 2. 12 ÷ 2 = __ Notice that we are multiplying to solve the division problem.

97. Dividing Fractions 9 ÷ 2 = __ Since 9 is one and one-half as much as 6, 9 ÷ 2 is one and one-half as much as 6 ÷ 2. So 9 ÷ 2 = 1 x (6 ÷ 2) = 1 x 3 = 4 . Let’s try 9 divided by 2. 1 2 1 2 1 2

98. Dividing Fractions Since 9 is one and one-half as much as 6, 9 ÷ 2 is one and one-half as much as 6 ÷ 2. So 9 ÷ 2 = 1 x (6 ÷ 2) = 1 x 3 = 4 . 9 ÷ 2 = __ 1 2 1 2 1 2

99. Dividing Fractions ÷ = 1 2 3 We’ll start by dividing 1 by various numbers. The last one can be thought of as how many 2/3s in 1 or half of 1 ÷ 1/3. 1 2 ÷ = 1 3 1

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

101. Dividing Fractions ÷ = 1 2 1 2 ÷ = 1 3 1 3 Each white pair is a reciprocal, sometimes called a multiplicative inverse. When multiplied together, they equal 1. In the equation 6 ÷ 2 = 3 , 2 x 3 = 6. 1 3 1 4 ÷ = 1 3 ÷ = 1 4 3 4 ÷ = 1 4 3 ÷ = 2 3 1 3 2

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

103. Dividing Fractions ÷ = __ 5 To find Use the same principle for dividing by a fraction except use 1 as the reference, rather than 6. 2 3

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

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

106. Dividing Fractions ÷ = __ To find (Is the answer more or less than 1?) Another example: How many 3/4s are in 2/3. More or less than 1? 2 3 3 4

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

108. Dividing Fractions ÷ = ÷ = 1 First find To find Then ÷ = __ The answer should be < 1 and it is. The extra step of dividing by 1 can later be omitted. 3 4 4 3 3 4 2 3 3 4 2 3 x ÷ ( 1 ) 3 4 2 3 = x = 4 3 8 9 2 3

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

110. Dividing Fractions It’s ours to reason why We invert and multiply. This presentation and handouts are available at ALabacus.com