Math Background

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 two-thirds, using this rectangle?
    Have a volunteer shade in two-thirds of the rectangle by drawing 2 parallel lines and shading in 2 smaller rectangles as shown below, left.
    rectangle
  • Say: Now I'm going to shade in three-fourths 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 three-fourths and two-thirds. The total number of little blocks in the diagram (12) is the product of the two denominators. The product of three-fourths x two-thirds is six-twelfths, or one-half.
  • Say: Note that our answer, one-half, is less than either three-fourths or two-thirds , since we multiplied two numbers less than one. You can think of three-fourths x two-thirds as three-fourths of two-thirds.
  • 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 five-ninths x three-tenths. 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:

    equation
  • 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, five-ninths x three-tenths = one-sixth. 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 three-tenths x two-thirds and four-sevenths x three-fourths. Have a volunteer explain how he or she solved the problems after the class has had time to solve them. (one-fifth and three-sevenths)
  • 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 3five-ninths x 1seven-eighths.
  • Say: In order to change 3five-ninths to an improper fraction, we need to write 3 as ninths by multiplying by one, like this. (Write three over one x nine-ninths = twenty-seven-ninths on the board and explain.) Therefore, 3five-ninths = twenty-seven-ninths + five-ninths = thirty-two-ninths.
    Write this on the board as you explain it.
  • Say: Similarly, 1seven-eighths = fifteen-eighths. (Show how to do this on the board.) Thus, 3five-ninths x 1one-eighth =
    equation

    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:

    2two-fifths x 1three-fourths and 4 x 1three-eighths. (4one-fifth and 5one-half)


Houghton Mifflin Math Grade 6