1. Presented by: Jenny Ray, Mathematics Specialist Kentucky Department of Education Northern KY Cooperative for Educational Services Jenny C. Ray Math Specialist, NKY Region Kentucky Department of Education1

2. Children’s Ideas about Fractions:  Show me where ½ could be on the number line below: Kentucky Department of Education2 0 1 2 3 Why do students sometimes choose this part of the number line?

3. Children’s Ideas about Whole Numbers:  3 > 2 ALWAYS.  1 = 1 ALWAYS.  So…how can it be that 1 / 3 > ½ ? Kentucky Department of Education3

4. When students can’t ‘remember’ a procedure, they resort to performing any operation they know they can do… Estimate the answer: 12 / 13 + 7 / 8 A) 1 B) 2 C) 19 D) 21 E) I don’t know. Kentucky Department of Education4

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6. National Assessment of Educational Progress (NAEP) results show an apparent lack of understanding of fractions by 9, 13, and 17 yr olds. Estimate the answer: 12 / 13 + 7 / 8  Only 24% of the 13-yr-olds responding chose the correct answer, “2”.  55% selected 19 or 21 These students seem to be operation on the fractions without any mental referents to aid their reasoning. Kentucky Department of Education6 Results from the 2 nd Mathematical Assessment of the National Assessment of Educational Progress

7. Perhaps you’ve seen this reasoning… 1 / 2 + 1 / 3 = 2 / 5 If students have an understanding of the value of the fractions on a number line, or as parts of a whole, then they can argue the unreasonableness of this answer. Kentucky Department of Education7

8. How can students learn to think quantitatively about fractions? Based on research… “…students should know something about the relative size of fractions. They should be able to order fractions with the same denominators or same numerators as well as to judge if a fraction is greater than or less than 1/2. They should know the equivalents of 1/2 and other familiar fractions. The acquisition of a quantitative understanding of fractions is based on students' experiences with physical models and on instruction that emphasizes meaning rather than procedures.” (Bezuk & Cramer, 1989) Kentucky Department of Education8

9. Hands on experiences help students develop a conceptual understanding of fractions’ numerical values. FRACTION MANIPULATIVES Kentucky Department of Education9

10. Learning Activity: Fraction Circles The white circle is 1. What is the value of each of these pieces? 1 yellow 3 reds 1 purple 3 greens Kentucky Department of Education10 Now…change the unit: The yellow piece is 1. What is the value of those pieces?

11. Learning Activity: Using Counters Eight counters equal 1, or 1 whole. What is the value of each set of counters?  1 counter  2 counters  4 counters  6 counters  12 counters Kentucky Department of Education11 Now, change the unit: Four counters equal 1. What is the value of each set of counters?

12. Learning Activity: Cuisinaire Rods The green Cuisenaire rod equals 1. What is the value of each of these rods?  red  black  white  dark green Kentucky Department of Education12 Change the unit: The dark green rod is 1. Now what is the value of those rods?

13. Learning Activity: Number Lines Kentucky Department of Education13

14. A “new” way of thinking/teaching…  “Many pairs of fractions can be compared without using a formal algorithm, such as finding a common denominator or changing each fraction to a decimal.” Kentucky Department of Education14

15. Comparing without an algorithm Pairs of fractions with like denominators: 1/4 and 3/4 3/5 and 4/5 Pairs of fractions with like numerators: 1/3 and 1/2 2/5 and 2/3 Pairs of fractions that are on opposite sides of 1/2 or 1: 3/7 and 5/9 3/11 and 11/3 Pairs of fractions that have the same number of pieces less than one whole: 2/3 and 3/4 3/5 and 6/8 Kentucky Department of Education15

16. Comparing 3/7 and 5/9…a student’s response: The fractions in the third category are on "opposite sides" of a comparison point. One fourth-grade student compared 3/7 and 5/9 in the following manner (Roberts 1985): "Three-sevenths is less. It doesn't cover half the unit. Five-ninths covers over half." Kentucky Department of Education16

17. Comparing 6/8 and 3/5: A student’s response… A fourth-grade student compared 6/8 and 3/5 in this way (Roberts 1985): "Six-eighths is greater. When you look at it, then you have six of them, and there'd be only two pieces left. And then if they're smaller pieces like, it wouldn't have very much space left in it, and it would cover up a lot more. Now here [3/5] the pieces are bigger, and if you have three of them you would still have two big ones left. So it would be less." Kentucky Department of Education17

18. Conceptual Understanding  Notice that each child's reasoning from the previous two examples is based on an internal image constructed for fractions.  Hands-on experiences with fractional parts, both smaller than and greater than one, helps to create this conceptual knowledge, so that procedures that they develop make sense. Kentucky Department of Education18

19. Exploring fractions with the same denominators Use circular pieces. The whole circle is the unit. – A. Show 1/4 – B. Show 3/4 Are the pieces the same size? How many pieces did you use to show 1/4? How many pieces did you use to show 3/4? Which fraction is larger? How do you know? Kentucky Department of Education19

20. Comparing fractions to ½ or 1 Use circular pieces. The whole circle is the unit. A. Show 2/3 B. Show 1/4 Which fraction covers more than one-half of the circle? Which fraction covers less than one-half of the circle? Which fraction is larger? How do you know? Compare these fraction pairs in the same way. – 2/8 and 3/5 – 1/3 and 5/6 – 3/4 and 2/3 Kentucky Department of Education20

21. Grade 3 Kentucky Department of Education21 Also MP6: Attending to Precision

22. Area Representations… Kentucky Department of Education22

23. Kentucky Department of Education23 “The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers.” Progressions for the Common Core; commoncoretools.org (2011)

24. 3.NF.2 Kentucky Department of Education24

25. 3.NF.3abc Kentucky Department of Education25

26. 3.NF.3d Kentucky Department of Education26

27. Grade 4  “Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction.” Progressions for the Common Core; commoncoretools.org (2011) Kentucky Department of Education27

28. 4.NF.1 Kentucky Department of Education28

29. 4.NF.2  Compare 5/8 and 7/12 Equivalent fractions..60/96 and 56/96  Compare 7/8 and 13/12 Use the benchmark of 1. Kentucky Department of Education29

30. 4.NF.3 Kentucky Department of Education30 Students in grade 4 are able to compose and decompose fractions using unit fractions and can write them as equations.

31. 4.NF.3c  Adding mixed numbers with like denominators by reasoning about the size of the numbers, not by using a “GPS” system…  7 ½ + 3 ½ =  2 ¾ + 5 ¾ = Kentucky Department of Education31

32. 4.NF.4  5 x 1/3 =  5 x 2/3 = Kentucky Department of Education32

33. 4.NF.5  Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add 2 fractions with respective denominators 10 and 100.  2/10 + 35/100 = Kentucky Department of Education33

34. 4.NF.6  Use decimal notation for fractions with denominators 10 or 100.  27/10 27/100  2.7 0.27 Kentucky Department of Education34

35. 4.NF.7 Kentucky Department of Education35

36. 5.NF.1 Kentucky Department of Education36

37. 5.NF.2 Kentucky Department of Education37

38. 5.NF.4b Kentucky Department of Education38

39. 5.NF.7  Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.  5 divided by ½… ½ divided by 3… Kentucky Department of Education39

40. Resources for Activities Illustrativemathematics.org  Illuminations (NCTM) Illuminations (NCTM)  Rational Number Project Rational Number Project  nzmaths nzmaths  Mars/Shell Centre Mars/Shell Centre  Teaching Channel Teaching Channel  www.jennyray.net www.jennyray.net Kentucky Department of Education40

41. Kentucky Department of Education41