
The ideas in this fraction lesson are taken from the Fractions 2 ebook sold on this website.
Only a few examples of each problem type are shown; you should make more problems of each kind for
the student.
Multiplying fractions by fractions
Free fraction lesson plan from HomeschoolMath.net
Most textbooks just plain 'announce' the rule for multiplying fractions by
fractions. This lesson will let you think and discover WHY the rule works. The several steps in this
lesson are all aiming towards something, so please follow them and do the exercises.
| We have studied how to find |
1

2 |
of a whole number. For example |
1

2 |
× 24 =
___. |
- Remember also that when you find a fraction of a number, the
word OF translates
into MULTIPLICATION.
|
| The same idea works when finding |
1

2 |
of a fraction! |
|
|
Example problem types
1. Suppose the pictures show how much pizza is left, and you share
it equally with your brother.
How much will you get? Write a multiplication sentence:
| Find |
1

2 |
of |
 |
1

2 |
× |
1

4 |
= |
|
| Find |
1

2 |
of |
 |
|
| Find |
1

2 |
of |
 |
|
|
Connection:
 |
1

2 |
of a number is the same as dividing the number by
2! |
|
|
|
1

2 |
× |
30 |
= and |
30 |
÷ 2 = |
|
|
1

2 |
× |
1

5 |
=
and |
1

5 |
÷ 2 = |
|
1

2 |
× |
1

7 |
= and |
1

7 |
÷ 2 = |
|
|
1

2 |
× |
1

10 |
=
and |
1

10 |
÷ 2 = |
|
 |
1

3 |
of a number is the same as dividing the number by 3! |
|
|
|
1

3 |
× |
30 |
= and |
30 |
÷3 = |
|
|
1

3 |
× |
1

2 |
=
and |
1

2 |
÷ 3 = |
|
 |
1

4 |
of a number is the same as dividing the number by 4! |
|
|
|
1

4 |
× |
28 |
= and |
28 |
÷ 4 = |
|
|
1

4 |
× |
1

2 |
=
and |
1

2 |
÷ 4 = |
|
|
| Remember? How to find |
3

4 |
of 16, or in other words |
3

4 |
× 16 ? You can think: |
|
1

4 |
of 16 is 4, so |
3

4 |
of 16 is 3 times as much. Therefore |
3

4 |
of 16 is
3 × 4 =12. |
Multiplying a fraction by a fraction means taking that fractional
part of the fraction.
It is just like taking certain part of the leftovers when
leftovers is already a fraction.
|
2. Suppose the pictures show how much pizza is left, and you get a
certain part of the leftovers.
How much will you get? Write a multiplication sentence. Color in an
answer picture.
Can you see how the multiplication means taking that fractional part of
the fraction?
It is just like taking certain part of the leftovers when
leftovers is already a fraction.
| First find |
1

4 |
of it, then multiply that by 3. |
1

4 |
× |
1

2 |
= |
1

8 |
, |
1

8 |
× 3
= |
|
|
| First find |
1

3 |
of it, then multiply that by 2. |
|
| First find |
1

4 |
of it, then multiply that by ___. |
|
| First find |
1

3 |
of it, then multiply that by ___. |
|
|
|
The rule for multiplying fractions is very easy:
Multiply the numerators and the
denominators separately.
|
For example: |
|
5

8 |
× |
3

4 |
= |
5 × 3

8 × 4 |
= |
15

32 |
|
|
|
|
|
|
|
3

7 |
× |
4

9 |
= |
3 × 4

7 × 9 |
= |
12

63 |
= |
4

21 |
|
|
|
3. Go back to exercise 2, and do the problems a-f using the rule.
Compare the results.
The following discussion justifies the rule by using an example.
| I want to find |
3

4 |
× |
5

7 |
, or if there is 5/7 of a pie left, how much is three fourths of
that. You can think: |
I can
first find 1/4 of the 5/7, or |
1

4 |
× |
5

7 |
, and take that 3 times. |
|
|
| |
|
1

4 |
× |
5

7 |
is the same as |
5

7 |
× |
1

4 |
, and that I can find doing |
1

7 |
× |
1

4 |
and taking that 5 times. |
|
|
1

7 |
× |
1

4 |
= |
1

28 |
. Taking that 5 times is 5 × |
1

28 |
= |
5

28 |
Taking previous result 3 times is 3 × |
5

28 |
= |
3 × 5

28 |
= |
15

28 |
. |
|
|
So the result is the same as if I had multiplied the numerators and
the denominators separately.
The same thinking process works with other fractions, too.
|
4. Multiply. Remember to always give your answer in lowest terms
(simplified) and as a mixed number, if possible.
| a. |
3

9 |
× |
2

9 |
|
|
| b. |
11

12 |
× |
1

6 |
|
|
| c. |
8 |
× |
3

13 |
|
|
| d. |
9 |
× |
2

3 |
|
|
| e. |
2

9 |
× |
8 |
|
5. Mary wants to make half of this recipe.
How much does she
need of each ingredient? |
| Brownies
3 cups sweetened carob chips
8 tablespoons Extra virgin olive oil
2 eggs
1/2 cup honey
1 teaspoon vanilla
3/4 cup whole wheat flour
3/4 teaspoon baking powder
1 cup walnuts or other nuts |
|

|
Find the missing factor. |
| × |
6

7 |
= |
1

7 |
|
|
| × |
1

4 |
= |
5

16 |
|
|
| × |
3

8 |
= |
1

16 |
|
|
| × |
2

5 |
= |
3

10 |
|
|
Multiplication as an area
The ideas in this fraction lesson are taken from the Fractions 2 ebook sold on this website.
Only a few examples of each problem type are shown; you should make more problems of each kind for the student.
|
FREE math homework help on www.PhatMath.com
-
40 Math forums for grades K-12 & College
-
Post/view math problems & solutions
- Blogs for students, parents, teachers
|

Over 150 hand-crafted quality math worksheets covering all 6th grade topics.
See our samples!
|
|