1. Teaching for Understanding: Fractions Dr. DeAnn Huinker, University of Wisconsin-Milwaukee Mathematics Teacher Leader (MTL) Seminar Milwaukee Public Schools February 2005

2. Teaching for Understanding: Fractions Session Goals To deepen knowledge of fractions and rational numbers. To reason with fraction benchmarks. To use conceptual thought patterns for comparing 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. What is a fraction? What is a rational number? Are they the same?

5. 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)

6. Solve. Represent your reasoning with diagrams, words, or symbols. Ms. Cook is rewarding 8 students for reaching their reading goals. She ordered 3 medium sized pizzas for them to share equally. How much pizza will each student get?

7. Write an equation Discuss and Justify What does each number in the equation represent? What operation is embedded in the situation? What rational number interpretation is illustrated? What are some common misconceptions or struggles or issues that this raises for you?

8. Estimate. More than or less than 1/2? More or less than 1 whole? 2 wholes? Share your reasoning with others. Consider how students might reason. 127 138 + is approximately?

9. 1 2 19 21 127 138 + is closest to: Consider why these are the choices and how a student might reason in selecting each response.

10. Benefits of Learning with Understanding All Read p. 6–7 #1&2:Motivating (p.7) Promotes More Understanding (p.8) #3&4:Helps Memory (p.9) Enhances Transfer #5&6:Influences Attitudes & Beliefs (p.10) Promotes Autonomous Learners

11. Understanding (1) Individually Read and mark assigned sections. (2) Pairs Identify 2–3 important ideas. (3) Table Small Group Pairs explain important ideas and why they were selected.

12. Discuss How might the ideas about understanding guide our thinking as we work with students, other teachers, administrators, and parents?

13. 1 2 19 21 127 138 + is closest to:

14. Examining Student Work Select a Facilitator. Each person gets 1 work sample. Review the work individually. Report to the Group: Summarize what is going. Comment on the knowledge the student is most likely drawing upon.

15. Examining Student Work What surprised, impressed, or concerned you?

16. MPS Mathematics Framework 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. Lack of proficiency results from pushing ahead within one strand but failing to connect what is being learned with other strands. (NRC, 2001)

17. 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?

18. 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

19. Comparison of Fractions Consider ways to reason with benchmarks when comparing these fractions. 5/7 or3/7 3/8 or 3/4 5/4 or 8/9 15/16 or 9/10 1 1/3 or 6/3

20. 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 strategy–Whats missing?)

21. Ordering Fractions on the Number Line (1) Deal out fraction cards (1-2 per person). (2) Allow quiet time to think about placements. (3) Taking turns, each person: Places one fraction on the number line, and Explains his/her reasoning using benchmarks and conceptual thought patterns.

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

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

24. Examining Student Work As you review the work, speculate on how the students might have been thinking about fractions and decimals. Relative amounts? Whole numbers? Benchmarks?

25. Walk Away Fractions as quantities. Benchmarks: 0, 1/2, 1, 2 Conceptual thought patterns. Turn to a person near you and share one idea that you are hanging on to from todays session.

26. References Behr, M., Lesh, R.. Post. T. & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.). Acquisition of mathematics concepts and processes. New York: Academic Press, 9-61. Lamon, S. J. (1999). Teaching fractions and ratios for understanding. Mahwah, NJ: Lawrence Erlbaum. National Research Council (NRC). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.