Multiplication and
Division of
Fractions
In reality, no one can teach mathematics.
Effective teachers are those who can
stimulate students to learn mathematics.
Educational research offers compelling
evidence that students learn mathematics
well only when they construct their own
mathematical understanding
Everybody Counts
National Research Council, 1989
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;

Understand the meanings for operations for whole
numbers.
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?
Multiplication of Fractions
Consider:
2 3

3 4

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?
Models for Reasoning
About Multiplication
Fraction of a fraction
 Linear/measurement
 Area/measurement models
 Cross Shading

We will think of multiplying fractions as
finding a fraction of another fraction.
2
3
How much is of ?
3
4
We use a fraction
square to represent
the fraction 34 .
Then, we shade
2
3
3
4
of
We can see that it is
6 .
the same as 12
3
4
2 3 1
 
3 4 2
The Linear Model with multiplication utilizes
the number line and partitions the fractions
3
4
2
3
How much is of ?
3
4
1
4
0
1
3
of
3
4
3
4
2
4
2
3
of
3
4
1
2
3
3
of
3
4
2 3 1
 
3 4 2
4
4
We can also use the linear model with
shapes and partition accordingly
Identify ¾ of the circle
2
3
How much is of ?
3
4
Take 2 pieces
Break into
3 pieces
Answer is ½
2 3 1
 
3 4 2
In the third method, we will think of
multiplying fractions as multiplying a
length times a length to get an area.
Length is
3
4
2
3
How much is of ?
3
4
2
3
Area
Width is
Number of square units
Is 6 out of 12
This area is 2
3
2 3 1
 
3 4 2
X
3 = 6
4
12
Modeling multiplication of fractions using
the length times length equals area
approach requires that the children
understand how to find the area of a
rectangle.
A great advantage to this approach is that
the area model is consistently used for
multiplication of whole numbers and
decimals. Its use for fractions, then is
merely an extension of previous
experience.
In the fourth method, we will represent both
fractions on the same square.
2
3
2
3
How much is of ?
3
4
2
is 3
3
4
3
is 4
2 3 1
 
3 4 2
Modeling multiplication of fractions using
the cross shading approach does
produce correct answers. However,
many elementary students may not
grasp the
“because it is shaded in both
directions”
overlapping concept. This may require
some additional explanations
Classroom Problem

Eric and his mom are making cupcakes.
Each cupcake gets 1/4 of a cup of frosting.
They are making 20 cupcakes. How much
frosting do they need?
Sample children’s strategies
1/4 of a cup
1 cup
2 cups
“…so 5 cups altogether.”
3 cups
4 cups
5 cups
Another student strategy
1/4 of a cup
So, 5, 6, 7, 8 -- that’s 2 cups.
9, 10, 11, 12 -- that’s 3 cups.
13, 14, 15, 16 -- that’s 4 cups.
4 of these is 1 cup…
17, 18, 19, 20 -- that’s 5 cups.
…so 5 cups altogether.
Another student strategy
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
5 cups
1/4 + 1/4 + 1/4 + 1/4 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
Q: What’s a number sentence for this problem?
A: 20 x 1/4 = 5 (there are others)
Other Contexts for
Multiplication of Fractions

Finding part of a part (a reason why
multiplication doesn’t always make things
“bigger”)

Pizza (pepperoni on ⅓ of ½ pizza)

Recipes ( 1¾ cups of sugar is used but we
want to make ½ a batch)

Ribbon (you have ⅜ yd , ⅓ of the ribbon is
used to make a bow)
Division With Fractions
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?
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?
Division of Fractions examples




How many quarters are in a dollar?
Ground beef cost 2.80 for ½ pound. What is the
price per pound?
Maggie can walk the 2 ½ miles to school in 3/4 of
an hour. How long would it take to walk 4 miles?
Barb had ¾ of a pizza left over from her party. She
wants to store it in plastic containers. Each
container holds ⅓ of a pizza. How many
containers will she use? How many will be
completely full? How full will the last container be?
Division of Fractions examples

1
2
You have 1 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?
“How many one eighths are in three
fourths?”
3 1
 ?
4 8
Our pizza is cut into 8
pieces. If three fourths of
a pizza is left, how many
slices remain?
Recall: a slice represents
one eighth of the pizza
Pizza
How many one eighths are in three fourths?
3 1
 ?
4 8
To find this we must first
find 3/4 of the pizza.
We then cut each fourth into
halves to make eighths.
We can see there are 6
eighths in three fourths.
3 1
 6
4 8
Pizza
1 1
 ?
2 8
Now only half of the pizza is
left. How many slices remain?
How many one eighths are in
one half?
Using a fraction
manipulative, we show
one half of a circle.
To find how many one eighths are in one half, we cover
the one half with eighths and count how many we use.
We find there are 4. There are four one eighths in one half.
1 1
 4
2 8
3
:
1
4
4
1
1
:
2
6
1
1
:
3
12
2
2
3
:
3
=
=
Now that you know
how to divide
fractions, let’s try
some together.
3
3
3
=
=
4
1
:
1
3
3
1
1
2
:
12
=
=
3
6