Reasoning with Rational Numbers (Fractions)

1. Reasoning with Rational Numbers (Fractions) DeAnn Huinker, Kevin McLeod, Bernard Rahming, Melissa Hedges, & Sharonda Harris, University of Wisconsin-Milwaukee Mathematics Teacher Leader (MTL) Seminar Milwaukee Public Schools March 2005 www.mmp.uwm.edu • This material is based upon work supported by the National Science Foundation Grant No. EHR-0314898.

2. Reasoning with Rational Numbers (Fractions) Session Goals • To deepen knowledge of rational number operations for addition and subtraction. • To reason with fraction benchmarks. • To examine “big mathematical ideas” of equivalence and algorithms.

3. 26 50150 13 13 What’s in common? 0.333333... 33 %

4. 26 13 13 Big Idea: Equivalence • Any number or quantity can be represented in different ways. For example, , , 0.333333..., 33 % all represent the same quantity. • Different representations of the same quantity are called “equivalent.”

5. Big Idea: Algorithms • What is an algorithm? • Describe what comes to mind when you think of the term “algorithm.”

6. 713 Benchmarks for “Rational Numbers” • Is it a small or big part of the whole unit? • How far away is it from a whole unit? • More than, less than, or equivalent to: • one whole? two wholes? • one half? • zero?

7. Conceptual Thought Patterns for Reasoning with 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–What’s missing?)

8. Task: Estimation with Benchmarks • Facilitator reveals one problem at a time. • Each individual silently estimates. • On the facilitator’s cue: • Thumbs up = greater than benchmark • Thumbs down = less than benchmark • Wavering “waffling” = unsure • Justify reasoning.

9. Rational Number vs Fraction • Rational Number = How much?Refers to a quantity,expressed with varied written symbols. • Fraction = NotationRefers to a particular type of symbol or numeral used to represent a rational number.

10. Characteristics ofProblem Solving Tasks • 1: Task focuses attention on the “mathematics” of the problem. • 2: Task is accessible to students. • 3: Task requires justification and explanation for answers or methods.

11. Characteristics ofProblem Solving Tasks • IndividuallyRead pp. 67-70, highlight key points. • Table GroupDesignate a recorder.Discuss characteristics & connect to task. • Whole GroupReport key points and task connections.

12. Discuss Identify benefits of using problem solving tasks: • for the teacher? • for the students?

13. 12 58 1– = Task • Write a word problem for this equation.In other words, situate this computation in a real life context.

14. 12 58 12 34 1– = + = Task • Write a word problem for each equation. • Draw a diagram to represent each word problem and that shows the solution. • Explain your reasoning for how you figured out each solution.

15. 13 15 13 15 13 15 Which is accurate? Why? 1 – = • Alexis has 1 yards of felt. She used of a yard of felt to make a costume. How much is remaining? • Alexis has 1 yards of felt. She used of it for making a costume. How much felt is remaining?

16. Notes for comparing the two fraction situations. Whole = 1 yard of felt1 1/5 yards of felt. Use 1/3 of a yard of felt to make a costume. 1 1/5 yards – 1/3 yards = 2/3 yards + 1/5 yards = 13/15 yards Whole = 1 1/5 yards of felt1 1/5 yards of felt. Use 1/3 of the whole piece of felt to make a costume. 6/5 yards – (1/3 x 6/5) = 6/5 yards – 2/5 yards = 4/5 yards

17. Examining Student Work • Establish two small groups per table. • Designate a recorder for each group. • Comment on accuracy and reasoning: • Word Problem • Representation (Diagram) • Solution

18. Summarize • Strengths and limitations in students’ knowledge. • Implications for instruction.

19. + = 34 12 NAEP Results: Percent Correct • Age 13 35% • Age 17 67% National Assessment of Education Progress (NAEP)

20. + = 34 12 MPS Results

21. Research Findings: Operations with Fractions • Students do not apply their understanding of the magnitude (or meaning) of fractions when they operate with them (Carpenter, Corbitt, Linquist, & Reys, 1981). • Estimation is useful and important when operating with fractions and these students are more successful (Bezuk & Bieck, 1993). • Students who can use and move between models for fraction operations are more likely to reason with fractions as quantities (Towsley, 1989). Source: Vermont Mathematics Partnership (funded by NSF (EHR-0227057) and US DOE (S366A020002))

22. Fraction Kit Fold paper strips • Purple: Whole strip • Green: Halves, Fourths, Eighths • Gold: Thirds, Sixths, Ninths, Twelfths

23. Representing Operations Envelope #1 • Pairs • Each pair gets one word problem. • Estimate solution with benchmarks. • Use the paper strips to represent and solve the problem. • Table Group • Take turns presenting your reasoning.

24. Representing Operations Envelope #2 • As you work through the problems in this envelope, identify ways the problems and your reasoning differ from envelope #1. • Pairs: Estimate. Solve with paper strips. • Table Group: Take turns presenting.

25. + = 34 56 14 1112 – = Representing Your Reasoning Using plain paper and markers,clearly represent your reasoning with diagrams, words, and/or symbols for:

26. Representing Operations Envelope #3 • Pairs • Each pair gets one reflection prompt. • Discuss and respond. • Table Group • Take turns, pairs facilitate a table group discussion of their prompt.

27. Big Idea: Algorithms • Algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.

28. Walk Away • Estimation with benchmarks. • Word problems for addition and subtraction with rational numbers. • Representing situations. Turn to a person near you and summarize one idea that you are hanging on to from today’s session.

30. Estimation Task Greater than or Less than • 4/7 + 5/8 Benchmark: 1 • 1 2/9 – 1/3 Benchmark: 1 • 1 4/7 + 1 5/8 Benchmark: 3 • 6/7 + 4/5 Benchmark: 2 • 6/7 – 4/5 Benchmark: 0 • 5/9 – 5/7 Benchmark: 0 • 4/10 + 1/17 Benchmark: 1/2 • 7/12 – 1/25 Benchmark: 1/2 • 6/13 + 1/5 Benchmark: 1/2

31. Word Problems: Envelope #1 • Alicia ran 3/4 of a marathon and Maurice ran 1/2 of the same marathon. Who ran farther and by how much? • Sean worked on the computer for 3 1/4 hours. Later, Sean talked to Sonya on the phone for 1 5/12 hours. How many hours did Sean use the computer and talk on the phone all together? • Katie had 11/12 yards of string. One-fourth of a yard of string was used to tie newspapers. How much of the yard is remaining? • Khadijah bought a roll of border to use for decorating her walls. She used 2/6 of the roll for one wall and 6/12 of the roll for another wall. How much of the roll did she use?

32. Word Problems: Envelope #2 • Elizabeth practices the piano for 3/4 of an hour on Monday and 5/6 of an hour on Wednesday. How many hours per week does Elizabeth practice the piano? • On Saturday Chris and DuShawn went to a strawberry farm to pick berries. Chris picked 2 3/4 pails and DuShawn picked 1 1/3 pails. Which boy picked more and by how much? • One-fourth of your grade is based on the final. Two-thirds of your grade is based on homework. If the rest of your grade is based on participation, how much is participation worth? • Dontae lives 1 5/6 miles from the mall. Corves lives 3/4 of a mile from the mall. How much closer is Corves to the mall?

33. Envelope #3. Reflection Prompts • Describe adjustments in your reasoning to solve problems in envelope #2 as compared to envelope #1. • Summarize your general strategy in using the paper strips (e.g., how did you begin, proceed, and conclude). • Describe ways to transform the problems in envelope #2 to be more like the problems in envelope #1. • Compare and contrast your approach in using the paper strips to the standard algorithm.