Multiplying & Dividing Fractions –

The Challenge of Computation vs. Conceptualization

Math Alliance July 27, 2010

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

Learning Intentions & Success Criteria

Learning Intention – We are learning to: Deepen our conceptual understanding of division and

multiplication with fractions.

Success Criteria – You will be able to: Use real-life problem-solving situations to surface the

meaning of division and multiplication of fractions.

Popcorn PartyServing Size: 2 cups

How many servings from:

8 cups of popcorn

5 cups of popcorn

SummarizePopcorn Party: Serving Size 2 cupsHow many servings from 8 cups; 5 cups?

Is this a measurement or partitive context?Write the equation for each situation

(8 cups; 5 cups).Identify the meaning of each number in the

equations.Draw a picture, make a diagram, or use paper

strips to show how you got your answer.

SummarizeHow many 2–cup servings?8 ÷ 2 = 4

cups cup servings serving

12

servings

5 ÷ 2 = 2 cups cup

serving

What does the represent?12

One-half of a cup of popcorn?

One-half of a serving? Are you sure?

Popcorn PartyServing Size: cup

How many servings from:

12

12

1 cup of popcorn

4 cups

2 cups

SummarizeHow many 2–cup servings?

1 ÷ = 2 cup servings

4 ÷ = 8 cups servings

12

cupservin

g12

cupservin

g

TaskPopcorn Parties #1 & #2

Facilitator

Paper Strip Demonstrator

Algorithm

TaskPopcorn Parties #1 & #2

Facilitator poses one problem at a time. Each individual silently solves it. On facilitator’s cue: State answer. Take turns as the demonstrator who

models with paper strips. Take turns to justify your reasoning.

Popcorn Party #1

Serving Size: cup of popcorn

How many servings can be made from: 1 cup of popcorn 2 cups of popcorn 3 1/2 cups of popcorn

14

Popcorn Party #2

Serving Size: cup of popcorn

How many servings can be made from: 3/4 cup of popcorn 6 cups of popcorn 2 1/4 cups of popcorn 4 1/2 cups of popcorn

34

Popcorn Party

Serving Size: cup

How many servings from 4 cups?

Using your paper strips, work individually to solve the problem.

THEN compare results.

34

4 ÷ = cups

34

cupservin

gJust as with whole numbers, it is important to understand the meaning of the answer and how to interpret and relate ways of reporting “remainders.”

Think about a “leftover or extra” amount

Think about the “number of groups”

Think about the “size of a group”

Interpreting RemaindersFor each problem, Place it in a context Solve it with paper strips Interpret the solution

with both a “leftover” and as it relates to the “groups.”

Think about “leftover or extra” amount the “number of groups” the “size of a group”

78

38

583 ÷ = ?

÷ = ?

Discussion

In reviewing these division situations and

your solutions, what are you noticing?

–œ ‡°± ·þ

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Consider ⅔ × ¾

As you crafted a real-world situation, what struggles emerged?

Describe a real-world situation that can be modeled by this equation.What could the ⅔ represent?What could the ¾ represent?

Turn and Share

Back to the basics…

Consider for a moment 2×3.What could the 2 represent?What could the 3 represent?

• What insights surfaced?

• In what ways are 2×3 and ⅔ × ¾ similar?

• In what ways are they different?

• Does the meaning of multiplication change when moving from whole numbers to fractions?

What do you understand about 2/3 x 3/4?

“Algorithms for multiplication of common fractions are easy for

teachers to teach and students to use, but their meanings are elusive.”

-- Zhijun Wu

Julie is making the family dinner. She buys 4 packages of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy?

Turn and talk – What does the 4 represent?What does the 5/8 represent?

Individually –•Work through this problem.•Record your thinking on note cards.•Represent your thinking using numbers, pictures, & words.•Place face down in the middle of your table when done.

Debriefing Strategies

(1) Each person picks a card to study.(2) After 30 seconds pass the card to the right.(3) Study the strategy on each card.(4) After all cards have been passed, share comments,

questions or ah-ha’s with the table group.

Table DiscussionHow do these strategies demonstrate an

understanding of multiplication?

Looking at student work...

In what ways do the students show understanding of the situation as multiplication? How do students show that they understand the “number of groups” and the “size of groups” in their representations?

Julie is making the family dinner. She buys 4 packages of meat for the spaghetti. Each package weighs 5/8 lb. How many pounds of meat does she buy?

“These types of situations can be modeled by the repeated-addition interpretation. The link between multiplication and addition is clearly seen here. The repeated-addition model offers a satisfying interpretation in this case.”

--Zhijun Wu

How does your thinking change when you consider this next problem . . .

Taking a run...

Put your pencils down. Turn and talk –

What does the 4 represent?

What does the 5/8 represent?

I wanted to run 4 miles. I ran 5/8 of the distance before I stopped for water. How many miles did I run before I had to stop for water?

Individually –•Work through this problem.•Record your thinking on note cards.•Represent your thinking using numbers, pictures, & words.•Place face down in the middle of your table when done.

Taking a run...Strategy Debrief1. Each person picks a card to study.2. After 30 seconds pass the card to the right.3. Study the strategy on each card.4. After all cards have been passed, share comments, questions or ah-ha’s with the table group.

Table DiscussionHow do these strategies demonstrate an understanding of multiplication?

Thinking about parts and wholes

Combining parts:Problem 1 4 = packages of meat

5/8 = weight per package (quantity)

“4 parts of 5/8 lbs each”

Finding part of a group:

Problem 2 4 = miles OR the whole run

5/8 = part of the 4 miles (5/8 is now the operator – we do not need a complete whole but we need a part of that whole.)

“5/8 parts of 4 miles”

In what ways do the students show understanding of the situation as multiplication?

Taking a Run – Looking at Student Work

What does multiplication of fractions encompass?

Multiplication of fractions involves:Combining equal parts Finding a part of a whole or part of a group Doing both – combining equal parts and

finding part of a whole.

Problem Sort•Read through each problem.•As a table, decide which problem type is represented in each context.•Solve using pictures, diagrams, and numbers.

From Whole Numbers to Fractions

What experiences do students need to extend the meaning of multiplication from whole numbers to multiplication of fractions? Experience with real-life problem solving

situations. Use of concrete and pictorial

representations to support students as they reason.

Opportunities to explore the meaning of multiplication through a variety of problem formats involving fractions.

HomeworkTo help you prepare for the exam next week it is recommended that you complete the following:

Division of FractionsRead Beckmann pp. 326-329Do Practice Problems for Section 7.4 pp. 335-339 #1, #3, #8a, b, c

Multiplication of FractionsRead Beckmann pp. 263-268Do Practice Problems for Section 6.1 pp.268-269

Problems for Card Sort Mrs. Smith has 120 books in her fourth grade classroom. 4/5 of the books are fiction. How many books are fiction? All notebooks at the local store are discounted by ¼ A notebook originally cost $0.96. How much do you save on one notebook if you buy it today? Julie bought 4/5 of a yard of fabric for her class project. Later she found that she needed only ¾ of the material. How much material did Julie use for her project? At the supermarket potatoes are bagged in ¾ pound bags. Mom bought 3 bags of potatoes. How many pounds of potatoes did mom buy? Red cabbage cost $0.39 a pound. Julie bought 3 1/3 pounds of red cabbage to prepare her dish. How much did she pay for the red cabbage? I put a container holding a half gallon of ice cream into the freezer. Two days later the ice cream container is 2/3 full. How much ice cream is in the container? Melissa is planning on making several batches of cookies. She needs 2/3 cup of sugar for every batch she makes. She plans on making 2 ½ batches. How much sugar will she need to make these batches.