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### 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.

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.

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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.”

Big Idea: Algorithms

- What is an algorithm?
- Describe what comes to mind when you think of the term “algorithm.”

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?

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?)

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.

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.

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.

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.

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Task

- Write a word problem for this equation.In other words, situate this computation in a real life context.

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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.

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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?

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

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

Summarize

- Strengths and limitations in students’ knowledge.
- Implications for instruction.

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12

NAEP Results: Percent Correct

- Age 13 35%
- Age 17 67%

National Assessment of Education Progress (NAEP)

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))

Fraction Kit

Fold paper strips

- Purple: Whole strip
- Green: Halves, Fourths, Eighths
- Gold: Thirds, Sixths, Ninths, Twelfths

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.

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.

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– =

Representing Your ReasoningUsing plain paper and markers,clearly represent your reasoning with diagrams, words, and/or symbols for:

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.

Big Idea: Algorithms

- Algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.

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.

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

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?

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?

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.

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