1. Developing Fraction Concepts Math Alliance July 13, 2010 Beth Schefelker, DeAnn Huinker, Chris Guthrie & Melissa Hedges

2. Learning Intentions and Success Criteria We are learning to… deepen our knowledge of fractions and rational numbers We will be successful when… we can apply knowledge of fraction benchmarks and conceptual thought patterns to reason and compare fractions.

3. 3434 How do students see this fraction? Students often see fractions as two whole numbers (Behr et al., 1983). What are ways we want students to “see” and “think about” fractions?

4. Different models offer different opportunities to learn. Area model – visualize part of the whole Use the grey triangles to cover ¾ of the octagon. Length or linear model – emphasizes that a fraction is a number as well as its relative size to other numbers 1 ½ 2 Where would ¾ fall on this number line? Why? Set Model – the whole is set of objects and subsets of the whole make up fractional parts. 3/4 of the smiley faces are blue

5. What is a fraction? What is a rational number? Are they the same?

6. Rational Number vs Fraction Rational Number = How much? Refers to a quantity or relative amount, expressed with varied written symbols. Fraction = Notation Refers to a symbol or numeral used to represent a rational number. (Lamon, 1999)‏

7. Reasoning About Fractions Name the fraction shown in the shaded region of the figure below: Share your responses. What do you notice? What do I need to consider as I decide on an answer?

8. Who’s right? Go to a poster that has a different answer than yours. Defend why that response could be correct. Now come up with a second argument to defend your answer. “The study of fractions offers many delightful and challenging opportunities to practice mathematical reasoning.” p. 65 Beckmann

9. 2 ¼ ¾ 9/12 9/4 9 Consider each response above as you respond to each question: What’s the whole? What are the parts? Big Ideas A fraction tells us the relationship between the part and the whole. A fraction is always a fraction of some whole. The whole needs to be understood while working with the fraction even if it is not made explicit. Models help clarify ideas and visualize the relationships between numerator and denominator.

10. What does the research say about how students use fractions? A majority of U.S. students have learned rules but understand very little about what quantities the symbols represent and consequently make frequent and nonsensical errors. (NRC, 2001)‏ Share one error you’ve seen your students make.

11. Reason with “Rational Numbers” and Use Benchmarks Is it a small part of the whole unit? Is it a big part? More than, less than, or equivalent: to one whole?to one half? Close to zero?

12. Finish these fractions so they are close to but greater than one-half. Finish these fractions so they are close to but less than 1 whole. 9 15 12 21 11 24 16 85

13. Comparison of Fractions Consider ways to reason with benchmarks when comparing these fractions. 5/7 or 3/7 3/8 or 3/4 5/4 or 8/9 15/16 or 9/10 1 1/3 or 6/3

14. Conceptual Thought Patterns for Comparing Fractions More of the same-size parts. Same number of parts but different sizes. More or less than one-half or one whole. Distance from one-half or one whole (residual piece).

15. Ordering Fractions on the Number Line Deal out fraction cards (1-2 per person). Allow quiet time to think about placements. Taking turns, each person: Places one fraction on the number line, and Explains his/her reasoning using benchmarks and conceptual thought patterns. Warning: No conversions to decimals! No common denominators! No cross multiplying!

16. Fraction Cards 3/83/106/5 7/477/10025/26 7/1513/2414/30 16/1711/95/3 8/317/12

18. Using Representations to Conceptualize Fractions How did you think about the fraction 8/3? What does 8/3 mean? Develop a real-life context for 8/3? Make a representation for your story that helps develop an understanding of 8/3.

19. Reflect As you placed the fractions on the number line, summarize some new reasoning or strengthened understandings.

20. Walk Away Fractions as quantities. Benchmarks: 0, 1/2, 1, 2 Conceptual thought patterns.

21. Homework Beckmann Read pp. 65-70 Class Activities: p. 33 1 and 4 Also recommended though not required “Practice Problems for Section 3.1” p. 70 #7 Using reasoning other than finding common denominators, cross-multiplying, or converting to decimal numbers to compare the sizes (greater than, equal to, or less than) of the following fractions: 1/49 1/39 7/37 7/35 13/25 5/8 17/18 19/20