Transcript Document

Multiplication of Fractions:
Thinking More Deeply
Steve Klass
48th Annual Fall Conference of the California Mathematics Council - South
Palm Springs, CA, Nov. 2, 2007
Today’s Session

Welcome and introductions

What students need to know well before
operations with fractions

Contexts for multiplication of fractions

Meanings for multiplication

Models for multiplication of fractions

Discussion
2
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.
3
A Context for Fraction
Multiplication

Nadine is baking brownies. In her family, some
people like their brownies frosted without
walnuts, others like them frosted with walnuts,
and some just like them plain.
So Nadine frosts 3/4 of her batch of brownies
and puts walnuts on 2/3 of the frosted part.
How much of her batch of brownies has both
frosting and walnuts?
4
Multiplication of Fractions
Consider:
2 3
3 2
 and 
3 4
4 3

How do you think a child might solve each of these?

Do both representations mean exactly the same thing
to children?

What kinds of reasoning and/or models might they use
to make sense of each of these problems?

Which one best represents Nadine’s brownie problem?
5
Reasoning About Multiplication

Whole number meanings - 2 U.S. textbook conventions

4x2=8
• Set - Four groups of two
• Area - Four rows of two columns
6
Reasoning About Multiplication

2x4=8
• Set - Two groups of four
• Area - Two rows of four columns
• When multiplying, each factor refers to something
different. One factor can tell how many groups there are
and the other, how many in each group. The end result is
the same product, but 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 a group of three-fourths of one whole
Area - Two-thirds of a row of three-fourths of one column
3 2 1
 
4 3 2
•
•
Set - Three-fourths of a group of two-thirds of one whole
Area - Three-fourths rows of two-thirds of one column
8
Models for Reasoning
About Multiplication

Area/measurement models
(e.g. fraction circles)

Linear/measurement (e.g paper tape)
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Materials for Modeling
Multiplication of Fractions
 How
could 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
• Paper folding
10
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
11
Using an Area Model with Fraction
Circles for Fraction Multiplication

How could you use these materials to model
2 3 1
 
3 4 2
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Materials for Modeling
Multiplication of Fractions
 How
could you use these materials to
model 3  2  1 ?
4
3
2
• Paper tape
• Fraction circles

You could also use:
• Pattern blocks
• Fraction Bars / Fraction Strips
• Paper folding
13
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
14
Using an Area Model with Fraction
Circles for Fraction Multiplication
 How
could you use these materials to
model 3  2  1 ?
4
3
2
15
Mixed Number Multiplication
1
1
How could you solve 2 Ê Ê3 ?
4
2

Using a ruler and card, draw a rectangle that is 2 1 by
1
3
2
4
inches, and find the total number of square inches.
Find your answer first by counting, then by multiplying.

Compare your answers, are they the same?
16
Mixed Number Multiplication

Can you find out what each square is worth?
 What about partial squares?
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Making Connections …
1
1
1
1
2 ÊÊ3  (2ÊÊ Ê)(3ÊÊ )
4
2
4
2
1
3 ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ
2
(2  3) =Ê6
1
(2  ) =Ê1
2
2
1 3
(3  ) =
4 4
1
4
1 1 1
(  )=
4 2 8
7
7
8
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Try it Yourself
 How
can you use these materials to
model
2 1 1
1 2 1
  ÊandÊ  
3 2 3
2 3 3
?
 What
contexts can you construct for
these two problems?
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Other Contexts for
Multiplication of Fractions

Finding part of a part (a reason why
multiplication doesn’t always make things
“bigger”)

2
3


Pizza (pepperoni on
Brownies
pecans)
1
(2
is
of
1
)
2
2
frosted, 3 of
1
2
Ribbon (you have yd ,
used to make a bow)
2
3
the that part has
of the ribbon is
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Thinking More Deeply About
Multiplication of Fractions

Estimating and judging the reasonableness of
answers

Recognizing situations involving multiplication
of fractions

Considering, creating and representing
contexts where the multiplication of fractions
occurs

Making careful number choices
21
Questions/Discussion
22
Contact Us
[email protected]
http://pdc.sdsu.edu
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