Transcript Year 1 Math Classes:

Multiplication and Division of
Fractions: Thinking More Deeply
Nadine Bezuk and Steve Klass
Session 322 CMC-S 2005
Today’s Session

Welcome and introductions
 Meanings for operations
• How do we help children model and reason about the
operations
• Multiplication and division with whole numbers
• Multiplication and division with fractions

Models for multiplication of fractions
• Set, Array, Area, Measurement

Models for division of fractions
• Set, Array, Area, Measurement

Questions
What Students Need to Know Well
Before Operating With Fractions

Meaning of the denominator (number of equal-sized
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 operations for whole
numbers.
Types of Models for
Considering Fractions

Area/region
•

Length/linear
•

Fraction circles, pattern blocks, paper folding,
geoboards, fraction bars, fraction strips/kits
Number lines, rulers, (fraction bars, fraction
strips/kits)
Set/discrete
•
Chips,counters, painted beans
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?

What Do Children Need to Know in Order to
Understand Division With Fractions?
QuickTime™ and a
Sorenson Video 3 decompressor
are needed to see this picture.
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?
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 much does each group get?
How does this work?
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?
Dealing With Remainders
1
1
2
1
3
1
0
1
1
3
1
3
1
3
1
3
?
1
2
A Context For Division of
Fractions

1
1 2 cups
You have
of sugar. It takes 31 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?
Models for Reasoning
About Division

Area/Measurement (batches of cookies)

Linear/Measurement (ribbon - external
units)

How do you deal with the remainder?
Materials for Modeling
Division of Fractions

How would you use these materials to model
1
1
2
1
?
3
• Paper tape
• Fraction circles

You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips
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?
An Eighth Grade Problem?
3
4
3
5
George has of a pie. He ate of that.
How much pie did he eat?

17% of 13-year-olds answered correctly,
though 60% could correctly calculate
7 3

the product
.
8 2
(NAEP, 1983, p. 26)
Another Context
At one school 3/4 of all eighth graders
went to one game. Two-thirds of those
who went to the game traveled by car.
What part of all the eighth graders
traveled by car to the game?

12% chose to multiply, while about 55%
decided to subtract and about 8% to
divide! (Sowder, 1988)
Reasoning About Multiplication

Whole number meanings - U.S. conventions
• 4x2=8
• Set - Four groups of two
• Array - Four rows of two columns
• Measurement - Four units by two units
• 2x4=8
• Set - Two groups of four
• Array - Two rows of four columns
• Measurement - Two units by four units
Reasoning About Multiplication

Fraction meanings - U.S. conventions
2 3 1
 
3 4 2
•
•
•
Set - Two-thirds of one group of three-fourths
Array - Two-thirds of a row of three-fourths of one column
Measurement - Two-thirds of one unit by three-fourths of one unit
3 2 1
 
4 3 2
•
•
•
Set - Three-fourths of one group of two-thirds
Array - Three-fourths rows of two-thirds of one column
Measurement - Three-fourths of one unit by two-thirds of one unit
Contexts for Multiplication

Finding part of a part (a reason why
multiplication doesn’t always make things
“bigger”)

Pizza (pepperoni on
3
( 4 is
frosted, 2 of the that part has

Brownies
pecans)

Lawn ( 3 is mowed,
4
2 3
of )
3 4
3
2
3
of that is raked)
Models for Reasoning
About Multiplication

Area/measurement models (fraction
circles)

Linear/measurement (ribbon)

Set models (eggs in cartons)
Materials for Modeling
Multiplication of Fractions

How would you use these materials to
model 2  3  1 ?
3
4
2
• Paper tape
• Fraction circles

You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips
Questions?
References
National Assessment of Educational Progress. (1983) The third
national mathematics assessment: Results, trends and issues.
Report No. 13-MA-01. Denver: Education Commission of the
States.
Sowder, L. (1988) Concept-driven strategies for solving story
problems in mathematics. Final Tech. Rep, Center for Research
in Mathematics and Science Education, San Diego State
University, San Diego, CA (ERIC Document Reproduction
Service No. ED 290 629)
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