Multiply and Divide Fractions

Area Model of Multiplication
Multiplication of two fractions can be modeled using area. Remind students that one way to understand the product of 3 and 4 is as the area of a rectangle that is 3 units by 4 units.

Similarly, × equals the area of a rectangle that is units by units. Draw this rectangle inside a unit square. Since the area of the whole square is 1 and it is made up of 3 × 4 = 12 equal rectangles, the area of each little rectangle is . There are 2 × 3 shaded rectangles, so the area of the shaded region is 2 × × 4 = = .

In general, a rectangle that is by can be separated into a × c non-overlapping rectangles, each with area × d. Thus, the area of the rectangle is a × × d.

Multiplication of Fractions
The multiplication rule for fractions is quite easy to remember: Multiply numerators and then multiply the denominators.

where b ≠ 0, d ≠ 0

The development of the rule for multiplying and actually requires two steps: multiplying by a and dividing by b. Multiplication by a counting number involves repeated addition.

Example:

The general rule is where d ≠ 0.

Division of a fraction by a counting number takes more thought. What is ÷ 3? Think of as two unit fractions of . Three does not divide into 2 evenly. However, we can write as 2 × × 3 = . This is 6 unit fractions of , and 3 divided by 3. So the quotient equals 2 unit fractions of , or .

The general rule is ÷ b = × d where b ≠ 0, d ≠ 0.

Notice that when d = 1, this shows that c ÷ b = . Using what has been shown above, × = a × × d can be derived. First divide the fraction by b and then multiply by a.

÷ b = × d and a × × d = a × × d.

Division of Fractions
The division rule is easy to state—multiply by the reciprocal—but it requires a careful explanation.

where b ≠ 0, c ≠ 0, d ≠ 0

The reciprocal of the fraction is the fraction . A number times its reciprocal equals 1.

Multiplication and division are inverse operations, so 6 ÷ 2 = 3 because 3 × 2 = 6. Similarly, divided by can be thought of as the number of fractions in (the solution to = m × ). By multiplying both sides of this equation by the reciprocal of , the solution is m = × . So ÷ = × .

Fractions can also be divided by using a common denominator. How many 6-inch ribbons can be cut from a ribbon that is 15 inches long? The answer can be found by dividing 15 by 6: 15 ÷ 6 = = = 2 ribbons. This same idea can be used to show the rule for division of fractions. Recall that the numerator of a fraction tells how many parts there are and the denominator tells the size of each part. How many s are there in 1? Find 1 ÷ . The problem 1 ÷ is the same as ÷ . Since the denominators are the same, divide the numerators. The answer is 7 ÷ 2 = = 3. The quotient of fractions with the same denominator is the quotient of the numerators.

Teaching Model 12.5: Divide Fractions