Houghton Mifflin Mathematics Grade 6: Chapter 4: Fractions and Mixed Numbers: Introducing the Concept


Adding and Subtracting Fractions and Mixed Numbers Introducing the Concept Developing the Concept Multiplying and Dividing Fractions and Mixed Numbers Introducing the Concept Developing the Concept

Multiplying and Dividing Fractions and Mixed Numbers

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 $times$ is , or .
Say: Note that our answer, , is less than either or , since we multiplied two numbers less than one. You can think of $times$ 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 $times$ . 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 $times$ 2, or 6, in the denominator. Thus, $times$ = . 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

$times$

and

$times$

. 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 $times$ 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