5.3.1 Division of Fractions

Number & Operations for Teachers

Modeling the Meaning of Division

When dividing whole numbers, indicated quotients such as 12 ¸ 2 are often interpreted as asking, “How many 2’s are there in 12?” or “How many times can you subtract 2 from 12?” Like division of whole numbers, division of fractions also may be thought of as repeated subtraction. For instance, the expression 2 ¸ 1/2 = 4 may be modeled as a measurement process in which the quantity 1/2 is repeatedly subtracted from 2 until zero is obtained. In this model of division, the quotient 4 is interpreted as the number of times the subtraction occurs. A word problem such as the following may help to justify this interpretation: Mary has 2 cups of sugar for making cookies. If each batch of cookies requires 1/2 cup of sugar, how many batches can she make? In this context, it is easy to see why a fraction operation involving the numbers 2 and 1/2 could produce an answer that is a whole number larger than either 2 or 1/2.

In the partitive model of division, the divisor represents the number of groups into which the dividend is partitioned. Consequently, the divisor must be a whole number. As a result, the expression 2 ¸ 1/2 has no obvious partitive interpretation.

Fraction bar, number line, and area models may be used to illustrate fraction division. For instance, Figures 5.8 and 5.9 use fraction bars and number lines as concept models for the operation 4/3 ¸ 1/3 = 4, or to answer the question “How may thirds are there in four-thirds?”

Figure 5.10 uses an area model to represent the operation 1/2 ¸ 1/6 = 3, or, “How many sixths are there in one-half?” In this model, each cell has area 1/6 that of the square unit. The horizontal and vertical cell divisions make it clear that one-half is equivalent to three-sixths. Therefore, 1/2 ¸ 1/6 = 3.

Figure 5.10: Area Model for 1/2 ¸ 1/6 = 3

“How many groups of 3/15 are there in 24/15?”

Figure 5.11: Area Model for 24/15 ¸ 3/15 = 8

Figure 5.11 models the operation 24/15 ¸ 3/15 = 8. Note that the quotient, 8, is interpreted as the number of groups of size 3/15 that are contained in 24/15.

Example 5.1

Create a concept model and related algorithm representing the operation 4/5 ¸ 1/3, or, “How many thirds are there in four-fifths?”

Solution 5.1

Since …

· 4/5 of the unit rectangle may be represented as the first four columns of cells, and

· 1/3 of the unit rectangle may be represented as a row of five cells,

The cells in the first four columns are shaded in groups of five. Doing so results in two groups of five plus two-fifths of a group.

The quotient 4/5 ¸ 1/3 may also be modeled using fraction bars. Note that the 2/5 of 1/3 corresponds to the quantity 2/15.

Example 5.2

Create a concept model and related algorithm representing the operation 1/3 ¸ 4/5.

Solution 5.2

Since …

· 1/3 of the unit rectangle may be represented as the first row of five cells, and

· 4/5 of the unit rectangle may be represented the first four columns of cells, or 12 cells

The five cells in the first row may be thought of as 5/12 of the cells in the first four columns. So, the quotient 5/12 represents the fraction of the first four columns contained in the first row.

The quotient 1/3 ¸ 4/5 may also be modeled using fraction bars. Using fifteenths as a common divisor, it is seen that one third is 5/12 of four-fifths.