Reasoning with Rational Numbers (Fractions)

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Reasoning with Rational Numbers (Fractions). DeAnn Huinker, Kevin McLeod, Bernard Rahming, Melissa Hedges, &amp; Sharonda Harris, University of Wisconsin-Milwaukee Mathematics Teacher Leader (MTL) Seminar Milwaukee Public Schools March 2005 www.mmp.uwm.edu.

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### Reasoning with Rational Numbers (Fractions)

DeAnn Huinker, Kevin McLeod, Bernard Rahming, Melissa Hedges, & Sharonda Harris,

University of Wisconsin-Milwaukee

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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50150

13

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What’s in common?

0.333333...

33 %

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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.”

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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?)
• 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.
• 1: Task focuses attention on the “mathematics” of the problem.
• 2: Task is accessible to students.
• IndividuallyRead pp. 67-70, highlight key points.
• Table GroupDesignate a recorder.Discuss characteristics & connect to task.
• Whole GroupReport key points and task connections.
Discuss

Identify benefits of using problem solving tasks:

• for the teacher?
• for the students?

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1– =

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

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1– =

+ =

• 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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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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Using 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.

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?