Lesson: Multiplying and Dividing Fractions and Mixed Numbers
Introducing the Concept
Showing students a visual model for the multiplication of fractions before you teach the algorithm may help some students gain a better understanding of multiplying fractions and mixed numbers.
Materials: paper and pencil
Preparation: none
Prerequisite Skills and Background: Students should be able to multiply whole numbers and have an understanding of the model that uses the area of a rectangle for the multiplication of two whole numbers. They also should be able to find the prime factorization of a number.
Draw a picture of a rectangle on the board.
- Ask: Who can come up to the board and show us
, using this rectangle?
Have a volunteer shade in of the rectangle by drawing 2 parallel lines and shading in 2 smaller rectangles as shown below, left.
- Say: Now I'm going to shade in
of the rectangle by drawing 3 parallel lines horizontally. (See the figure on the right.) In the diagram the number of little rectangles that are shaded twice (6) is the product of the numerators of and . The total number of little blocks in the diagram (12) is the product of the two denominators. The product of x is 
, or 
. - Say: Note that our answer, , is less than either or , since we multiplied two numbers less than one. You can think of x as of .
- Say: Who can tell us what we do when we want to multiply two fractions?
Multiply the numerators and then multiply the denominators. Simplify the fraction, if possible. - Say: Good; now let's try an example together. Let's look at multiplying
x 
. Instead of multiplying first, it is simpler if we write the numbers in their prime factorization forms and cancel any common factors.
Write this on the board:
- Say: Note that we can cancel the common factors of 3 and 5 since they are in both the numerator and denominator, leaving 1 in the numerator and 3 x 2, or 6, in the denominator. Thus, x =
. Canceling the common factors before multiplying makes simplifying a lot easier. - Say: Now try these at your desk and I'll come around to help those who need it.
On the board, write x and 
x . Have a volunteer explain how he or she solved the problems after the class has had time to solve them. (
and 
) - Say: When multiplying mixed numbers, the first thing we need to do is to write the mixed numbers as improper fractions and multiply the two improper fractions. Let's look at 3 x 1
. - Say: In order to change 3 to an improper fraction, we need to write 3 as ninths by multiplying by one, like this. (Write
x 
= 
on the board and explain.) Therefore,
3 = + = 
.
Write this on the board as you explain it. - Say: Similarly, 1 =
. (Show how to do this on the board.) Thus, 3 x 1
=
Write the equations on the board as you explain.
- Say: Try a couple of these at your desks.
Write these on the board for students to do:2
x 1 and 4 x 1
.
(4 and 5)