Modeling Multiplication and Division
of Fractions
So what’s new about fractions in
Grades 6-8?
SOL 6.4
The student will demonstrate multiple
representations of multiplication and division
of fractions.
2
Thinking About Multiplication
The
expression…
We read it…
It means…
It looks like…
23
1
2
3
1 1

2 3
3
Thinking About Multiplication
The
expression…
23
1
2
3
1 1

2 3
4
We read it…
It means…
2 times 3
two groups of
three
2 times
1
3
1 times 1
2
3
two groups of
one-third
one-half group
of one-third
It looks like…
Making sense of multiplication of
fractions using paper folding and
area models
• Enhanced Scope and Sequence, 2004,
pages 22 - 24
5
The Importance of Context
• Builds meaning for operations
• Develops understanding of and helps
illustrate the relationships among operations
• Allows for a variety of approaches to solving
a problem
6
Contexts for Modeling Multiplication
of Fractions
The Andersons had pizza for dinner, and there was
one-half of a pizza left over. Their three boys each
ate one-third of the leftovers for a late night snack.
How much of the original pizza did each boy get for
snack?
7
1 1
1
 
3 2
6
One-third of one-half of a pizza is equal to one-sixth of a pizza.
Which
meaning of
multiplication
does this
model fit?
8
Another Context for
Multiplication of Fractions
• Mrs. Jones has 24 gold stickers that she
bought to put on perfect test papers. She
1
took 2 of the stickers out of the package, and
then she used 1 of that half on the papers.
3
• What fraction of the 24 stickers did she use on
the perfect test papers?
9
1 1
1
 
3 2
6
One-third of one-half of the 24 stickers is
1
6
of the 24 stickers.
What
meaning(s) of
multiplication
does this
model fit?
Problems involving discrete items
may be represented with set models.
10
What’s the relationship between
multiplying and dividing?
• Multiplication and division are inverse relations
• One operation undoes the other
• Division by a number yields the same result as
multiplication by its reciprocal (inverse). For example:
1
62  6
2
11
Meanings of Division
For 20 ÷ 5 = 4…
Divvy Up (Partitive): “Sally has 20 cookies. How many
cookies can she give to each of her five friends, if she gives
each friend the same number of cookies?
- Known number of groups, unknown group size
Measure Out (Quotitive): “Sally has 20 minutes left on
her cell phone plan this month. How many more 5-minute calls
can she make this month?
- Known group size, unknown number of groups
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA
Publishing, 1998.
12
Sometimes, Always, Never?
• When we multiply, the product is larger than the
number we start with.
• When we divide, the quotient is smaller than the
number we start with.
13
“I thought times makes it bigger...”
When moving beyond whole numbers to situations involving
fractions and mixed numbers as factors, divisors, and
dividends, students can easily become confused. Helping
them match problems to everyday situations can help them
better understand what it means to multiply and divide with
fractions. However, repeated addition and array meanings of
multiplication, as well as a divvy up meaning of division, no
longer make as much sense as they did when describing
whole number operations.
Using a Groups-Of interpretation of multiplication and a
Measure Out interpretation of division can help:
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA
Publishing, 1998.
14
“Groups of” and “Measure Out”
1/4 x 8: “I have one-fourth of a box of 8 doughnuts.”
8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are
there?”
1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what
fraction of a gallon of gas do I have in my tank?”
1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans
do I have?”
3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½
laps?
3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be
easier to describe using divvy up.
2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?”
3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a
pie?”
2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you
make if you have 2½ cups of oil left?”
15
Thinking About Division
The
expression…
20 ÷ 5
20 
16
1
2
We read it…
It means…
It looks like…
Thinking About Division
The
expression…
20 ÷ 5
We read it…
20 divided
by 5
It means…
It looks like…
20 divided into
groups of 5;
20 divided into 5
equal groups…
How many 5’s are
in 20?
1
20 
2
20 divided
by 1
2
20 divided into
groups of 1 …
2
How many 1 ’s are
2
in 20?
17
17
Thinking About Division
The expression…
1 1

2 3
We read it…
one-half
divided by
one-third
It means…
It looks like…
1
2
divided into
groups of 1 …
3
How many 1 ’s are
3
1
in 2 ?
Is the quotient more than one
or less than one? How do you
know?
18
?
Contexts for
Division of Fractions
The Andersons had half of a pizza left after
1
dinner. Their son’s typical serving size is 3
pizza. How many of these servings will he eat
if he finishes the pizza?
19
1 1
1
 1
2 3
2
1
1
1
2 pizza divided into 3 pizza servings = 1 2 servings
1 serving
1
2 serving
20
Another Context for
Division of Fractions
1
3
Marcy is baking brownies. Her recipe calls for
cup
cocoa for each batch of brownies. Once she gets
1
started, Marcy realizes she only has 2 cup cocoa. If
Marcy uses all of the cocoa, how many batches of
brownies can she bake?
21
1 1
1
 1
2 3
2
1 cup
Three batches (or
Two batches (or
1
2
cup
3
3
cup)
2
3
cup)
1
1
2
One batch (or 13 cup)
0 cups
22
batches
Another Context for
Division of Fractions
1
2
Mrs. Smith had of a sheet cake left over after
her party. She decides to divide the rest of the
1
cake into portions that equal 3 of the original
cake.
1
3
How many cake portions can Mrs. Smith make
from her left-over cake?
23
What could it look like?
1 1

2 3
24
What does it look
like numerically?
25
What about
the traditional algorithm?
• If the traditional “invert and multiply” algorithm is taught,
it is important that students have the opportunity to
consider why it works.
• Representations of a pictorial nature provide a visual for
finding the reciprocal amount in a given situation.
• The common denominator method is a different, valid
algorithm. Again, it is important that students have the
opportunity to consider why it works.
26
What about
the traditional algorithm?
Later, think about divisors with numerators > 1.
Think about 1 ÷
2
.
3
2
How many times could we take 3 from 1?
1
We can take it out once, and we’d have 3 left. We could only take half
2
3
of another 3 from the remaining portion. That’s a total of 2 .
2
3
In each unit, there are sets of .
3
2
27
Multiple Representations
Instructional programs from pre-k through grade 12
should enable all students to –
• Create and use representations to organize,
record and communicate mathematical ideas;
• Select, apply, and translate among mathematical
representations to solve problems;
• Use representations to model and interpret
physical, social, and mathematical phenomena.
from Principles and Standards for School Mathematics (NCTM, 2000), p. 67.
28
Using multiple representations to
express understanding
Given problem
Contextual situation
Check your solution
Solve numerically
29
Solve graphically
Using multiple
representations
to express
understanding of
division of
fractions
30