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.

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.

a.      3 4  ×    =   First find  1 4  of it, then multiply that by 3. 1 4  ×  1 2   =   1 8 ,           1 8   × 3 =  b.      2 3  ×     =   First find  1 3  of it, then multiply that by 2. c.      3 4  ×    =   First find  1 4  of it, then multiply that by ___.   d.      2 3  ×    =   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.

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.

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