Making Sense of Adding and Subtracting Fractions

January MTL MeetingJanuary 20 & 28

Marshall Complex

Core Content Development TeamDeAnn HuinkerConnie LaughlinKevin McLeodMary MooneyBeth SchefelkerMelissa Hedges

Assisting MTLsFrelesha LeFloreCecile Labecki

Learning Intentions:

Develop “operation sense” related to adding and subtracting common fractions.

Understand how estimation should be an integral part of fraction computation development.

Success Criteria:At the end of this session you will be able to:

Justify your thinking when adding and subtracting fractions using concrete models and estimation strategies.

Types of Rational Numbers:

Part-whole comparisons

Quotients Operators Ratios and Rates Measurement

No pencils please!

For each expression below… (1) Estimate the sum and the difference for the expressions below.(2) Justify how you know your estimate is in the ballpark.

No computing please!

49/99 – 11/34 16/25 + 9/14

Background on Fraction Computation

Read page 160 only.

What ideas from the reading support the work we’ve been engaged in during the past 2 sessions around fractions?

From:Van de Walle, J. & Lovin H. (2006). Teaching Student Center

Mathematics: Grades 3-5. pp. 160-161. New York: Pearson Education.

Three Models Used to Work With Fractions

Set model (Mariah's Kittens)

Linear model (Taking a Run)

1 mile 2 miles 3 miles 4 miles Finding ⅝ of 4 miles

Area Model (Spaghetti Problem) ⅝ ⅝ ⅝ ⅝

4 packages of meat each weighing ⅝ lbs

Folding Fraction Strips

Fold fraction strips to represent the following:

one whole halves fourths

eighths thirds sixths

As you fold, discuss with your partner how you know that your fraction strips are accurate.

Folding fraction strips is so time consuming. Why do it?

Turn and talk: What was the benefit of

folding your own fraction strips?

In what way would the cognitive demand be affected if we were to hand you a pre-printed set of strips?

Exploring Operation Sense Using Benchmark Fractions

Using your fraction strips, work through the following and have several examples for each:

If you add 2 fractions and the sum is greater than ½, what can you say about the fractions?

If you add 2 fractions and the sum is greater than 1, what can you say about the fractions?

If you add 2 fractions, and the sum is more than 1 ½, what can you say about the fraction?

What conversations did you have with your partner that helped to develop a generalization about all three contexts?

Making connections to whole number operation sense…In what ways does this reasoning mirror the thinking we are asking children to develop as they work with whole numbers?

If you add 2 whole numbers together and the sum is larger than 10, what do you

know about those 2 numbers? What about 100? What about 150?

A firm understanding of fractions is the most critical foundation for fraction computation. Without this foundation, students will most certainly be learning rules without reasons.

---Van de Walle, p. 160

Exploring EqualitiesDivide into 2 teams.Team #1 – Task ATeam #2 – Task BUse your fraction strips as you work through this exploration.

Task A

½ ¾ 2/3 1 1 ¼ 1 1/3 1 ½

1.Choose a target number from this set that is less than 1. (Task B – More than 1)

2. Make a model of that number using your fraction strips to serve as a reference.

3.Using your fraction strips find combinations to represent your target number.

4.Start with combinations of 2 fractions, then 3, then 4, etc.

5.Use post-its to keep track of combinations you try. Each combination on it’s own post-it.

If you feel that you have explored all combinations for your first target number, pick a second one.

Example for Task A: Target Number ‹1

Target Number: ¾

Reference strip ¾

½ ¼ Model combinations of fractions that equal your target number:

Fraction strips folded and placed underneath the reference strip.

Record your thinking:

½ + ¼ = ¾

Equalities: Task A & Task B Debrief

What did you understand about fractions that allowed you to make combinations?

What patterns emerged as you made your combinations?

What big ideas from our work with whole numbers transfer to this work with fractions?

What ideas about addition of fractions surface as you engaged in these explorations?

How did the fraction strip model support your thinking?

Exploring DifferenceWorking with your shoulder partner:1. Decide…Which is larger? About by how much?2. Estimate the difference.3. Report your answer as a unit fraction.

No common denominators please!No pencils!

2/3 or 1/4

3/8 or 3/4

5/3 or 7/4

Exploring Difference

What strategy did you use to find the difference?

In what way is this similar to experiences students have as they explore subtraction of whole numbers?

What are we learning about the idea of difference?

Van de Walle & Lovin p. 160-161

The Dangerous Rush to Rules

andProblem-Based Number

Sense Approach

What challenges does this reading surface as we begin to think about building number sense and fraction algorithms with students?

What strategies do your teachers use to help address these challenges?

February Homework:

1. Please select a grade level that you will be studying next month. Bring a complete set of Teacher’s Editions for that specific grade level. Grade levels we will study include grades 2 – 9.

2. Please review the Revised WI Model Academic Standards

Grades 3 – 5 (Number and Operations Sense with Fractions)

6 – 8 Number Operations High School

Bring a copy to the February meeting.

Website: dpi.wi.gov/cal/math-intro.html

Task A

½ ¾ 2/3 1 1 ¼ 1 1/3 1 ½

1.Choose a target number from this set that is less than 1. (Task B – More than 1)

2.Make a model of that number using your fraction strips to serve as a reference.

3.Using your fraction strips find combinations to represent your target number.

4.Start with combinations of 2 fractions, then 3, then 4, etc.

5. Use post-its to keep track of combinations you try. Each combination on it’s own post-it.

Task B

½ ¾ 2/3 1 1 ¼ 1 1/3 1 ½

1.Choose a target number from this set that is greater than 1.

2. Make a model of that number using your fraction strips to serve as a reference.

3.Using your fraction strips find combinations to represent your target number.

4.Start with combinations of 2 fractions, then 3, then 4, etc.

5.Use post-its to keep track of combinations you try. Each combination on it’s own post-it.