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
?
14
Using an Area Model With a
Measurement Interpretation
1
1
Representation of 1
with
fraction
2
3
circles.
15
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
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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.
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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
22
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)
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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
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