Reasoning with Rational Numbers (Fractions)

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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.
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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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Whats in common?
33

• 0.333333...

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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.

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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?

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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
  strategyWhats missing?)

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Task Estimation with Benchmarks

• Facilitator reveals one problem at a time.
• Each individual silently estimates.
• On the facilitators cue
• Thumbs up greater than benchmark
• Thumbs down less than benchmark
• Wavering waffling unsure
• Justify reasoning.

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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.

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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.

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Characteristics ofProblem Solving Tasks

• Individually Read 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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Discuss

• Identify benefits of using problem solving
  tasks
• for the teacher?
• for the students?

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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?
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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?

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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
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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

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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)

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MPS Results
Grade 5 Grade 6 Grade 7 Grade 8 Overall
n 51 86 53 60 250
Correct Solution 51 25 30 30 33
Correct Word Problem 39 24 23 28 28
Accurate Diagram 37 15 21 20 22
ClearReasoning 24 15 34 15 21
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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))
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Fraction Kit

• Fold paper strips
• Purple Whole strip
• Green Halves, Fourths, Eighths
• Gold Thirds, Sixths, Ninths, Twelfths

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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.

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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 Reasoning

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

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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.

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Big Idea Algorithms

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

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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 todays session.
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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

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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?

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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?

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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.