Transcript Document

Multiplication and Division of Fractions:
Helping Increase the Certainty of
Understanding
Steve Klass and Nadine Bezuk
NCTM Annual Conference, Salt Lake City Utah, April 2008
Today’s Session

Welcome and introductions

Meanings for division and multiplication

Models for division and multiplication of
fractions

Contexts for division and multiplication
of fractions

Discussion
2
What Students Need to Know Well
Before Operating With Fractions

Meaning of the denominator (number of equalsized pieces into which the whole has been cut);

Meaning of the numerator (how many pieces are
being considered);

The more pieces a whole is divided into, the
smaller the size of the pieces;

Fractions aren’t just between zero and one, they
live between all the numbers on the number line;

A fraction can have many different names;

Understand the meanings for whole number
operations
3
Solving a Division Problem With
Fractions

How would you solve

How would you solve 11  1 ?
1 31 ?
2
3

How might a fifth or sixth grader solve these
problems and what answers might you
expect?

How can pictures or models be used to
solve these problems?
4
What Does Elliot Know?

What does Elliot understand?

What concepts is he struggling with?

How could we help him understand
how to model and reason about the
problem?
5
What Do Children Need to Know in Order to
Understand Division With Fractions?
6
What Does Elliot Know?



What does Elliot understand?
What concepts is he struggling with?
How could we help him understand how to
model and reason about the problem?
7
Reasoning About Division

Whole number meanings for division
6÷2=3
• Sharing / partitive
• What does the 2 mean? What does the 3 mean?
• Repeated subtraction / measurement
• Now what does the 2 mean and what does the 3
mean?
8
Now Consider 6
1
÷2

What does this mean?

How can it be modeled?

What contexts make sense for
– Sharing interpretation
– Repeated subtraction interpretation
9
Reasoning About
Division With Fractions
10
Reasoning About Division
With Fractions

Sharing meaning for division:
1
1 3
• One shared by one-third of a group?
• How many in the whole group?
• How does this work?
11
Reasoning About Division
With Fractions

Repeated subtraction / measurement meaning
1
1 3
• How many times can one-third be subtracted
from one?
• How many one-thirds are contained in one?
• How does this work?
• How might you deal with anything that’s left?
12
Materials for Modeling
Division of Fractions

How would you use these materials to
model
1
12

1
?
3
• Paper strips
• Fraction circles

You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips/ Paper tape 13
Using a Linear Model With a
Measurement Interpretation
1
12

1
3
How many one-thirds are in one and one-half?
1
0
1
1
3
1
3
1
3
1
3
1
2
?
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Using an Area Model With a
Measurement Interpretation
1
1
 Representation of 1 
with
fraction
2
3
circles.
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How Many Thirds?
?
1
2
0
1
3
1
3
1
1
1
3
12
1
3
?
1
3
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A Context For Division of
Fractions
 You
1
12
1
3
have
cups of sugar. It takes
cup to make 1 batch of cookies. How
many batches of cookies can you
make?
• How many cups of sugar are left?
• How many batches of cookies could
be made with the sugar that’s left?
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Reasoning About
Multiplication With Fractions
18
Multiplication of Fractions
Consider:
2 3

3 4
3 2

4 3

How do you think a child might solve each
of these?

What kinds of reasoning and/or models
might they use to make sense of each of
these problems?
19
Reasoning About Multiplication

Whole number meanings - U.S. conventions
• 4x2=8
• Set - Four groups of two
• Area - Four units by two units
20
Reasoning About Multiplication

Whole number meanings - U.S. conventions
• 2x4=8
• Set - Two groups of four
• Area - Two units by four units
• When multiplying, each factor refers to
something different. One tells how many groups
and the other, how many in each group. The
representations are quite different.
21
Reasoning About Multiplication

Fraction meanings - U.S. conventions
2 3 1
 
3 4 2
•
•
Set - Two-thirds of one group of three-fourths
Area - Two-thirds of three-fourths of a unit
3 2 1
 
4 3 2
•
•
Set - Three-fourths of one group of two-thirds
Area - Three-fourths of two-thirds of a unit
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Models for Reasoning
About Multiplication

Area/measurement models (fraction
circles)

Linear/measurement (e.g. paper strips)
23
Materials for Modeling
Multiplication of Fractions
 How
would you use these materials
to model 2  3  1 ?
3
4
2
• Paper strips
• Fraction circles
 You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips
• Paper folding/ paper tape
24
Using a Linear Model With
Multiplication
2
3
How much is of ?
3
4
1
4
0
1
3
Êof Ê
3
4
3
4
2
4
2
3
Êof Ê
3
4
1
2
4
4
3
3
Êof Ê
3
4
2
3
2
1
SoÊ ÊofÊ Êof 1Êis ÊorÊ .
3
4
4
2
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Using an Area Model with Fraction
Circles for Fraction Multiplication

How would you use these materials to model
2 3 1
 
3 4 2
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Using a Linear Model With
Multiplication
3
2
How much is of ?
4
3
1
2
1
3
0
1
2
Êof Ê
4
3
2
2
ÊofÊ
4
3
3
2
ÊofÊ
4
3
2
3
3
3
4
2
ÊofÊ
4
3
3
2
1
SoÊ ÊofÊ Êof 1ÊisÊ .
4
3
2
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Using an Area Model with Fraction
Circles for Fraction Multiplication
 How
would you use these materials to
model 3  2  1 ?
4
3
2
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Contexts for Multiplication

Finding part of a part (a reason why
multiplication doesn’t always make things
“bigger”)
2 3
 Pizza (pepperoni on of )
3 4
3
2
 Brownies ( is frosted, of the
4
3
that part has
pecans)

Lawn (
3
4
is mowed,
2
3
of that is raked)
29
Thinking More Deeply About
Multiplication and Division of Fractions

Estimating and judging the reasonableness of
answers

Recognizing situations involving multiplication
or division of fractions

Considering and creating other contexts
where the multiplication or division of fractions
occurs

Making thoughtful number choices when
considering examples
30
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