Transcript Powerpoint

Division of Fractions:
Thinking More Deeply
Steve Klass
National Council of Teachers of Mathematics
Kansas City Regional Conference, October 25, 2007
Southern California Fires
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Today’s Session

Welcome and introductions

What students should know before operating
with fractions

Watching a student use division with
fractions

Reasoning about division

Models for division of fractions

Contexts for division of fractions

Questions
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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
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Solving a Division Problem With
Fractions

How would you solve 1 1 ?
3

How would you solve 11  1 ?

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?
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3
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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?
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What Do Children Need to Know in Order to
Understand Division With Fractions?
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What Does Elliot Know?
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


What does Elliot understand?
What concepts is he struggling with?
How could we help him understand how to
model and reason about the problem?
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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?
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Now Consider 6 ÷
1
2

What does this mean?

What does the answer mean?

How could the problem be modeled?

What contexts make sense for
– Sharing interpretation
– Repeated subtraction interpretation
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Reasoning About
Division With Fractions
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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?
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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?
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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
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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?
?
0
1
1
3
1
3
1
3
1
3
?
1
3
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Contexts for Division
With Fractions
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A Context For Division of Fractions
 You
1
12
have
cups of sugar. It takes
cup to make 1 batch of cookies.
1
3
 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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Another Context For Division of Fractions
 You
1
2
have 1 yards of licorice rope. It
2
takes 3 yard to make one package of
licorice.
 How many packages can be made?
 How much of a yard of licorice is left?
 How much of the original amount of licorice is
left?
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Model Using Your Materials

Use your materials to model
1
1
2
2
3
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Thinking More Deeply About Contexts
for Division of Fractions

Which contexts work for division of fractions?

What aspects allow some contexts to work
better than others?

Develop your own new context for the
1 2
1
problem we just modeled,  .
2
3
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Thinking More Deeply About
Division of Fractions

Estimating and judging the reasonableness
of answers

Recognizing situations involving division of
fractions

Considering and creating contexts where the
division of fractions occurs

Using a reasoning approach to consider why
“invert and multiply” works
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Questions/Discussion
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© 2007 Professional Development Collaborative
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