Reasoning with Rational Numbers

(Fractions)

DeAnn Huinker, Kevin McLeod, Bernard Rahming,

Melissa Hedges, & Sharonda Harris, University of Wisconsin-Milwaukee

Mathematics Teacher Leader (MTL) SeminarMilwaukee 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.

What’s in common?

0.333333...

13

26

13

50150

33 %

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.”

13

26

13

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?

713

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 benchmarkThumbs down = less than benchmarkWavering “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

Individually Read 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?

Task

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

1

2

58

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.

3

4

1

2+ =

1

2

58

1 – =

Which is accurate? Why?

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?

15

13

1 – =

13

151

3

15

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

Notes for comparing the two fraction situations.

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.

NAEP Results: Percent Correct

Age 13 35%

Age 17 67%

National Assessment of Education Progress (NAEP)

3

4

1

2+ =

MPS Results3

4

1

2+ =

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%

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

PairsEach pair gets one word problem.Estimate solution with benchmarks.Use the paper strips to represent and solve the problem.

Table GroupTake 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.

Representing Your ReasoningUsing plain paper and markers,clearly represent your reasoning with diagrams, words, and/or symbols for:1112

1

4– =

3

4

5

6+ =

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.