Math Background

Lesson: Multiplying and Dividing Fractions and Mixed Numbers
Developing the Concept

Students can relate their understanding of the multiplication of fractions to the division of fractions and mixed numbers. Having students estimate and discuss answers will also help them gain a better understanding of the division of fractions.

Materials: paper and pencil

Preparation: none

Prerequisite Skills and Background: Students should be able to multiply and divide whole numbers. They should be able to multiply two fractions and be able to find the prime factorization for a number.

  • Say: I'd like you to multiply the following fractions: three-fifths x five-thirds, two-sevenths x seven-halves, and five-fourths x four-fifths.
    Write the problems on the board for students to do at their desks.
  • Ask: What did you notice about the answer to each problem? (Each answer equals 1.)
  • Say: When two numbers are multiplied and their product is 1, the two numbers are said to be reciprocals of each other. What is the reciprocal of
    five-ninths x three-halves x 4 x 2one-third x (nine-fifths, two-thirds, one-fourth, three-sevenths )
  • Say: To find the reciprocal of a fraction, we invert the fraction. This means that the numerator and denominator change places.
  • Say: When we want to divide two fractions, we find the reciprocal of the divisor and multiply the other fraction by it. Whenever possible, we will cancel common factors ahead of time to make the simplification process easier. Let's work on a problem together.

    On the board, write three-fourths × two-thirds.

  • Say: We solve this problem by finding the reciprocal of the divisor and rewriting the problem as a multiplication problem like this.

    Write three-fourths × two-thirds = three-fourths x three-halves

  • Say: Now if we multiply three-fourths x three-halves, we get nine-eighths = 1one-eighth.
  • Say: Now let's have you try a problem at your desks. If you need help, raise your hand and I'll come around and help you.

    On the board, write three-fifths ÷ seven-tenths. After the students have had some time to do the problem, ask for a volunteer to do the problem on the board. (six-sevenths

  • Say: When we want to multiply mixed numbers, what do we need to do first? (Change them to improper fractions.) We do the same if we want to divide mixed numbers. We first change them to improper fractions. Then we find the reciprocal of the divisor and multiply. Let's try this one.

    Write this problem on the board: 2four-fifths ÷ 2one-tenth.

  • Say: What did we say we should do first in this problem? (Write the mixed numbers as improper fractions.) Good, so 2four-fifths = fourteen-fifths and 2one-tenth = twenty-one-tenths, so
    equation
  • Say: Notice how we wrote the numbers as a product of prime factors so that we could cancel any common factors. Now you try one at your desks.

    Write the following problem on the board: 3three-fifths ÷ 1four-fifths. After giving the class time to solve the problem, have a student come to the board to explain how to do the division. (2)

  • Say: Let's try two more problems at your desks.

    Write the following on the board: 2 one-half ÷ 1 three-eighths and 2 three-fourths ÷ 4 two-fifths.
    Have students volunteer to explain how they solved the problems. (1nine-elevenths, five-eighths)

  • Ask: What are the steps we need to follow in order to divide two fractions or mixed numbers?

    Students will probably make a list like this one.

    1. Change any mixed numbers to improper fractions.
    2. Find the reciprocal of the divisor and set up to multiply the two fractions.
    3. Factor each number in the numerators and denominators and cancel any common factors.
    4. Multiply the numbers in the numerator and denominator and simplify the product, if possible.

Wrap-Up and Assessment Hints
Having students estimate the product or quotient of two fractions before adding or subtracting will help them recognize unreasonable answers. For example, if they divide two-thirds by one-fourth, they should recognize that division is counting how many one-fourths are in two-thirds. Since two-thirds > one-half, which is equivalent to two-fourths, there are at least 2 one-fourths in two-thirds. Also, since two-thirds < three-fourths, there are not 3 one fourths in two-thirds. So the answer lies between 2 and 3.

Have students do pattern activities such as the following and then write on a bulletin board some of the things they discover. These activities will help them see patterns in multiplication, such as the following.

multiplication problems

Ask what happened to the answer each time? (It was cut in half.)
Thus, the answer here must be 3. So what happens when we multiply a number by a fraction less than one? (The answer is less than the other number.)
To challenge some of your best students, ask them open-ended questions such as, “If the product of two fractions is three-eighths when simplified, what could those two fractions be?” (one-half x three-fourths, three-fifths x five-eighths, and so on) and “If the quotient of two fractions is 2, what could those two fractions be?” (one-half ÷ one-fourth, two-thirds ÷ one-third, and so on)

Houghton Mifflin Math Grade 6