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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.

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reasoning with rational numbers fractions

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 fractions1
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.
what s in common

26

50150

13

13

What’s in common?

0.333333...

33 %

big idea equivalence

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.”
big idea algorithms
Big Idea: Algorithms
  • What is an algorithm?
  • Describe what comes to mind when you think of the term “algorithm.”
benchmarks for rational numbers

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?
conceptual thought patterns for reasoning with fractions
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
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 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 of problem solving tasks
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 of problem solving tasks1
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.
discuss
Discuss

Identify benefits of using problem solving tasks:

  • for the teacher?
  • for the students?
slide13

12

58

1– =

Task

  • Write a word problem for this equation.In other words, situate this computation in a real life context.
slide14

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.
which is accurate why

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?
slide16

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
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
Summarize
  • Strengths and limitations in students’ knowledge.
  • Implications for instruction.
slide19

+ =

34

12

NAEP Results: Percent Correct

  • Age 13 35%
  • Age 17 67%

National Assessment of Education Progress (NAEP)

research findings operations with fractions
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
Fraction Kit

Fold paper strips

  • Purple: Whole strip
  • Green: Halves, Fourths, Eighths
  • Gold: Thirds, Sixths, Ninths, Twelfths
representing operations envelope 1
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
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.
representing your reasoning

+ =

34

56

14

1112

– =

Representing Your Reasoning

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

representing operations envelope 3
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 algorithms1
Big Idea: Algorithms
  • Algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.
walk away
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
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
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
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
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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