1. Making Sense of Fractions Juli K. Dixon, Ph.D. University of Central Florida

2. Perspective… A student said this…

3. When asked to compare 4/5 and 2/3, a student said, “I know that 4/5 is greater than 2/3.” Perspective…

4. A student said this… When asked to compare 4/5 and 2/3, a student said, “I know that 4/5 is greater than 2/3.” How would you respond? Perspective…

5. The student said… I made both fractions using manipulatives. I knew that 4/5 was bigger because 4/5 has 4 pieces and 2/3 only has 2 pieces and since 4 is greater than 2 then 4/5 is greater than 2/3. Perspective…

6. Would you ask this student to compare 22/23 and 26/27? Perspective…

7. Tell which Fraction is Greater 1. 1.3/7 and 5/8 2. 2.4/7 and 4/9 3. 3.9/10 and 5/4 4. 4.3/8 and 5/8 5. 5.6/7 and 8/9

8. Think about this… Alex and Jessica are racing their bicycles. Alex is 3/7 of the way to the finish line and Jessica is 2/3 of the way to the finish line. Which racer is closer to the finish line? How do you know?

9. Think about this… Marc and Larry each bought the same type of energy bar. Marc has 1/8 of his energy bar left, Larry has 1/10 of his energy bar left. Who has more energy bar left? How do you know?

10. Think about this… Riley and Paige each bought a small pizza. Riley ate 5/6 of her pizza, and Paige ate 7/8 of her pizza. Who ate more pizza? How do you know?

11. NOW Tell which Fraction is Greater 1. 1.3/7 and 5/8 2. 2.4/7 and 4/9 3. 3.9/10 and 5/4 4. 4.3/8 and 5/8 5. 5.6/7 and 8/9

12. A new perspective… Would you ask a student to compare 22/23 and 26/27?

13. Apply What You Know Order the following fractions from greatest to least: 4/5, 9/8, 5/11, 4/7, 5/6

14. Why Fractions?

15. Because sometimes they’re the only way to get your fair share…

16. Why Fractions? Because sometimes they’re the only way to get your fair share… This is particularly important when it comes to cookies and candy bars :)

17. Why Fractions? Because sometimes they’re the only way to get your fair share… This is particularly important when it comes to cookies and candy bars :) And the doorbell rang…

18. Share 2 cookies among 4 people.

19. Share 4 cookies among 3 people.

20. Share 4 cookies among 5 people.

21. Share 3 candy bars among 6 people.

22. How much of a candy bar will each person need to give the newcomer if a 7th person comes along?

23. Now you try… Consider this…

24. Engaging Students in Reasoning and Sense Making Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get?

25. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? Engaging Students in Reasoning and Sense Making

26. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? Solving this wouldn’t require much perseverance… but what if we said… Engaging Students in Reasoning and Sense Making

27. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

28. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

29. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

30. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

31. Consider this… A student is asked to share 4 cookies equally among 5 friends. How much of a cookie should each friend get? – Give each person the biggest unbroken piece of cookie possible to start. Engaging Students in Reasoning and Sense Making

32. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

33. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

34. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

35. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

36. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

37. Consider this… So how much of a cookie would person A get? Engaging Students in Reasoning and Sense Making

38. Consider this… So how much of a cookie would person A get? - How much is this all together? Engaging Students in Reasoning and Sense Making

39. Consider this… What is important here is that the problem requires diligence to solve and yet with perseverance the solution is within reach. Students are reasoning… Engaging Students in Reasoning and Sense Making

40. How do we support this empowerment? “… a lack of understanding [of mathematical content] effectively prevents a student from engaging in the mathematical practices” (CCSS, 2010, p. 8).

41. How do we support this empowerment? “… a lack of understanding [of mathematical content] effectively prevents a student from engaging in the mathematical practices” (CCSS, 2010, p. 8). When and how do we develop this understanding?

42. Consider this 2 nd grade class.

44. How might the Geoboard help develop this concept?

45. Use the geoboard to find all the possible ways of dividing a geoboard into fourths where the largest area on the geoboard is the whole. Record your answers on the geoboard dot paper.

46. Consider this 2 nd grade class. How might this student’s earlier experiences have influenced her understanding?

47. How do fractions progress through the standards?

48. Grade 1 -Students partition circles and rectangles into halves and fourths (also referred to as quarters – but money is not introduced until grade 2). (1.MD.1.2) - Students understand that decomposing wholes into more equal shares creates smaller shares. (1.G.1.3)

49. How do fractions progress through the standards? Grade 2 -Students partition circles and rectangles into halves, thirds, and fourths (describe whole as two halves, three thirds, four fourths). (2.G.1.3) - Students understand that equal shares does not mean equal shape (2.G.1.3)

50. How do fractions progress through the standards? Grade 3 -Students make sense of unit fractions and fraction symbols. (3.NF.1.1) – relate to linear measurement from grade 1 -Students use number line to make sense of fractions (3.NF.1.2) -Students model equivalent fractions including fractions equivalent to 1 (3.NF.1.2)

51. How do fractions progress through the standards? Grade 3 -Students make sense of comparing fractions using common numerator and common denominator strategies but not strategies based on benchmark fractions.(3.NF.1.2)

52. How do fractions progress through the standards? Grade 4 -Students develop rules for fraction equivalence and explain them using visual models. (4.NF.1.1) -Students recognize and generate equivalent fractions. (4.NF.1.1)

53. How do fractions progress through the standards? Grade 4 -Students compare fractions with different numbers by creating common numerator or denominator or using benchmark of 1/2. (4.NF.1.2) -Students justify comparisons. (4.NF.1.2)

54. How do fractions progress through the standards? Grade 4 -Students decompose fractions. (4.NF.2.3a) -Students add and subtract fractions and mixed numbers with like denominators. (4.NF.2.3b) -Students solve word problems with fraction addition/subtraction. (4.NF.2.3c)

55. How do fractions progress through the standards? Grade 4 -Students apply what they know about multiplying whole numbers to multiply a fraction by a whole number. (4.NF.2.4a/b) -Students solve word problems involving multiplying a fraction by a whole number. (4.NF.2.4c)

56. How do fractions progress through the standards? Grade 5 -Students add and subtract fractions with unlike denominators by finding common denominators. (5.NF.1.1) -Students solve word problems involving adding and subtracting fractions and check for reasonableness of responses using estimates based on benchmarks. (5.NF.1.2)

57. How do fractions progress through the standards? Grade 5 -Students multiply fractions by fractions. (5.NF.1.1) -Students solve word problems involving multiplying fractions and check for reasonableness of responses using estimates based on benchmarks. (5.NF.1.2)

58. How do fractions progress through the standards? Grade 5 -Students interpret fractions as division of numerator by denominator. (5.NF.2.3) -Students apply understanding of multiplication to multiply fractions or whole numbers by fractions. (5.NF.2.4) Students solve word problems involving multiplying fractions and mixed numbers using models or equations. (5.NF.2.6)

59. How do fractions progress through the standards? Grade 5 -Students apply understanding of division to divide unit fractions by whole numbers and whole numbers by unit fractions including problems situated in real world contexts. (5.NF.2.7)

60. Consider this 5 th grade class.

62. What was the misconception?

63. The whole is important…

64. Consider telling the “whole” story with pattern blocks. The whole is important…

65. Consider telling the “whole” story with pattern blocks. The whole is important… Use the yellow hexagon as the whole.

66. Consider telling the “whole” story with pattern blocks. The whole is important… Use the yellow hexagon as the whole. What fraction is represented by 5 green triangles?

67. Consider telling the “whole” story with pattern blocks. The whole is important… Use the yellow hexagon as the whole. What fraction is represented by 1 blue rhombus?

68. Now use 2 yellow hexagons as the whole. The whole is important…

69. Now use 2 yellow hexagons as the whole. The whole is important… What fraction is represented by 4 blue rhombuses?

70. Now use 1 red trapezoid and 1 blue rhombus combined as the whole. The whole is important…

71. Now use 1 red trapezoid and 1 blue rhombus combined as the whole. The whole is important… What fraction is represented by 2 red trapezoids?

72. Stories should be told in more than one way. The whole is important…

73. Stories should be told in more than one way. The whole is important… Consider using two-color counters to tell your story beginning at the end.

74. Determine the whole given the parts. The whole is important…

75. Determine the whole given the parts. The whole is important… If 6 counters represent 2/3 of the whole set, how many counters are in the entire set?

76. Determine the whole given the parts. The whole is important… If 8 counters represent 4/5 of the whole set, how many counters are in the entire set?

77. Determine the whole given the parts. The whole is important… If 10 counters represent 2/9 of the whole set, how many counters are in the entire set?

78. Picture this…

79. How can we help students to make sense of fractions and fraction operations? Picture this…

80. How can we help students to make sense of fractions and fraction operations? Picture this… Through estimation!

81. When asked the following question, only 24% of 13-year olds and only 37% of 17 year olds could estimate correctly. Consider this concerning data… Estimate 12/13 + 7/8. a)1b) 2 c) 19d) 21

82. Consider the highly technical paper plate… How do we address this?

83. Consider the highly technical paper plate… How do we address this? Show me 1/2

84. Consider the highly technical paper plate… How do we address this? Show me less than 1/2

85. Consider the highly technical paper plate… How do we address this? Show me more than 1/2

86. Consider the highly technical paper plate… How do we address this? What other fraction can you show me?

87. Consider the highly technical paper plate… How do we address this? What fraction should I show you?

88. Consider the highly technical paper plate… How do we address this? Can we use this for decimals?

89. Estimate the following:

90. 1/2 + 2/5 Estimate the following:

91. 2/6 + 3/11 Estimate the following:

92. 2 1/13 + 6/7 Estimate the following:

93. 3 4/5 + 1 1/3 Estimate the following:

94. 1 7/8 - 1/2 Estimate the following:

95. Just for fun…

96. 1/2 x 3/4 Estimate the following:

97. 1 2/3 x 3/5 Estimate the following:

98. 7/8 x 9 Estimate the following:

99. 5 1/4 ÷ 8/9 Estimate the following:

100. 5/6 ÷ 1/4 Estimate the following:

101. Jim ate 1/4 of a medium cheese pizza then he at 1/8 of a medium pepperoni pizza. What fractional part of a pizza did Jim eat? Let’s get back to stories…

102. Thomasenia had 2 1/2 yards of ribbon. She gave 2/3 yard of her ribbon to Matt. How much ribbon did Thomasenia have left? Let’s get back to stories…

103. Now it’s your turn to tell the story…

104. Write a story to support 3/4 + 5/8. Now it’s your turn to tell the story…

105. Write a story to support 4/5 - 1/2. Now it’s your turn to tell the story…

106. Write a story to support 1 1/6 + 2/3. Now it’s your turn to tell the story…

107. We’ve just begun to tell the story about teaching fractions with depth.

108. How will it end? We’ve just begun to tell the story about teaching fractions with depth.