Transcript Multiplying and Dividing Fractions – The Challenge of

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 Party
Serving Size: 2 cups
How many servings from:

8 cups of popcorn

5 cups of popcorn
Summarize
Popcorn Party: Serving Size 2 cups
How 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.
Summarize
8
÷ 2
=
4
How
many
2–cup
servings?
cups
cup
servings
serving
5
÷ 2
=
cups
cup
serving
1
2 2
servings
1
What does the 2
represent?

One-half of a cup of popcorn?

One-half of a serving?

Are you sure?
Popcorn Party
Serving Size: 12 cup
How many servings from:
1 cup of popcorn
4 cups
1
2 2 cups
Summarize
1
How
many
1
÷ 2 2–cup
=
2
cup
servings
servings?
cup
serving
4
÷
cups
1
2
cup
serving
=
8
servings
Task
Popcorn Parties #1 & #2
 Facilitator
 Paper Strip Demonstrator
Algorithm
Task
Popcorn 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
1
Serving Size:
cup of popcorn
4
How many servings can be made from:

1 cup of popcorn

2 cups of popcorn

3 1/2 cups of popcorn
Popcorn Party #2
3
Serving Size:
cup of popcorn
4
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
Popcorn Party
3
Serving Size: 4 cup
How many servings from 4 cups?
Using your paper strips, work
individually to solve the problem.
THEN compare results.
3
4
4
÷
cups
cup
serving
=
Just 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 Remainders
For 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”
3
7
÷
8
8
= ?
5
3 ÷ 8
= ?
Discussion
In reviewing these division
situations and your solutions,
what are you noticing?
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Consider ⅔ × ¾
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
As you crafted a real-world situation, what
struggles emerged?
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)
(2)
(3)
(4)
Each person picks a card to study.
After 30 seconds pass the card to the right.
Study the strategy on each card.
After all cards have been passed, share comments,
questions or ah-ha’s with the table group.
Table Discussion
How do these strategies demonstrate an
understanding of multiplication?
Looking at student work...
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?
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?
“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...
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?
Put your pencils down. 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.
Taking a run...
Strategy Debrief
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 Discussion
How 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”
Taking a Run –
Looking at Student Work
In what ways do the students
show understanding of the
situation as multiplication?
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.
Homework
To help you prepare for the exam next week it is
recommended that you complete the following:
Division of Fractions
Read Beckmann pp. 326-329
Do Practice Problems for Section 7.4 pp. 335-339 #1, #3,
#8a, b, c
Multiplication of Fractions
Read Beckmann pp. 263-268
Do 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.