Teaching Models

Multiply and Divide Fractions

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

[a over b] times [c over d] example where b ≠ 0, d ≠ 0

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


General rule

The general rule is a × c over d = a times c over d where d ≠ 0.

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

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

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

c over d ÷ b = c over b × a and a × c over b × d = a × c over b × d.

Area Model of Multiplication

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, two-thirds × three-fourths equals the area of a rectangle that is two-thirds units by three-fourths 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 one-twelfth. There are 2 × 3 shaded rectangles, so the area of the shaded region is 2 × three-thirds × 4 = six-twelfths = one-half.

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

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

a/b divided by c/d where b ≠ 0, c ≠ 0, d ≠ 0

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

c/d X d/c

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

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 = fifteen-sixths = five-halves = 2 one-half 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 one-halfs are there in 1three-fourths? Find 1three-fourths ÷ one-half. The problem 1three-fourths ÷ one-half is the same as seven-fourths ÷ two-fourths. Since the denominators are the same, divide the numerators. The answer is 7 ÷ 2 = seven-halves = 3one-half. The quotient of fractions with the same denominator is the quotient of the numerators.

Division by common denominator

Teaching Model 6.1: Estimate Fraction Products and Quotients

Houghton Mifflin Math Grade 6