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. Resources for Activities  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 Education21

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