Reasoning with Rational Numbers

(Fractions)

Math Alliance ProjectJuly 20, 2010

DeAnn Huinker, Chris Guthrie, Melissa Hedges,& Beth Schefelker,

Learning Intentions & Success Criteria

Learning Intentions: We are learning to…Develop “operation sense” related to adding and subtracting common fractions through contextual situations.Understand how estimation should be an integral part of fraction computation development.

Success Criteria: You will be able to…Justify your thinking when adding and subtracting fractions using concrete models and estimation strategies.

What’s in common?

0.333333...

13

26

13

50150

33 %

Big Idea: Equivalence

Any number or quantity can be represented in different ways.

Example, these all represent the same quantity.

0.333333… 33 %

Different representations of the same quantity are called “equivalent.”

13

26

1 3

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

8/15 or 11/15

7/20 or 7/9

6/10 or 9/5

11/12 or 7/8

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 whole or one-half (residual strategy–What’s missing?)

12 713 8

+ =Estimate

• 1

• 2

• 19

• 21

• Don’t Know

NAEP 13 yr

7%

24%

28%

27%

14%

National Assessment of Education Progress (NAEP); MPS n=72)

MPS 6-7-8

13%

9%

23%

41%

9%

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.

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)

Fraction Kit

Fold paper strips. Only mark the folds, no words or symbols on the strips.

Yellow: Whole strip

Purple: Halves, Fourths, Eighths

Green: 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 Group

Take turns presenting your reasoning.

Repeat

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.

Repeat

Representing Operations Envelope #3

Individual

Each person gets a reflection prompt.

Consider your thoughts on it.

Table Group

Take turns to facilitate a table group discussion of your prompt.

Representing Your Reasoning

Pose a word problem for each equation (context).

On paper, clearly represent your reasoning steps with diagrams, words, and symbols.

1112

14– =

34

56+ =

Big Idea: Algorithms

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

Adding or subtracting fractions may involve renaming or trading fractions until the parts are the same size. Sometimes one fraction needs renaming, sometimes both, and sometimes neither.

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.

Homework

If you would like more practice on placing fractions on the number line, Class Activity 3F (p. 49) is recommended.

Required homework Due July 27:Class Activity 3G: Equivalent Fractions (p. 41)

Justify using conceptual thought patterns and your fraction strips if the sum or difference for the expressions below are Greater than or Less than the indicated benchmark:

1/8 + 4/5 Greater than or Less than 15/6 + 7/8 Greater than or Less than 1 ½11/12 – 1/3 Greater than or Less than ½ 6/4 – 5/3 Greater than or Less than 0

Pose a word problem for each problem below. Next use your fraction strips to solve it, then clearly represent your reasoning with diagrams, pictures, words, and symbols:

11/12 - ¼ = _____¾ + 5/6 = ______

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 a yard of string 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.