Fraction Division from a Measurement Model

To help students learn to divide fractions from a measurement perspective by reasoning about pictures, and to record their explanations in writing.

Because there are two interpretations for whole number division, namely measurement and sharing, there are also two interpretations for fraction division. In this section, we discuss measurement division.

Fortunately, the measurement model can be applied directly to fraction division without modification. For example, suppose our division problem is 3/4 ÷ 1/3. Using the measurement model of division, we would interpret this number sentence as asking the question, How many 1/3s are in 3/4? The following picture, where we overlap 3/4 with a picture of a whole divided into thirds, suggests that there are at least two 1/3s in 3/4, with a small amount leftover:

How much is left over? By finding a common partitioning of twelfths, we get a better idea of how much is leftover:

The heavy lines show how the thirds match up with the smaller partitions in the 3/4. Note that there is one small piece leftover. There are two ways of viewing this leftover part: we can either interpret it in terms of the whole, or in terms of 1/3. In terms of the whole, the leftover part is 1/12, because it takes twelve pieces of this size to make a whole. However, this leftover part is also 1/4 of a 1/3, because it takes four pieces of this size to make a 1/3.

The problem, then, is knowing which name to use for the leftover part. Because the question we are trying to answer is, How many 1/3s are in 3/4, we usually write the answer as 2 1/4, meaning that there are 2 1/4 thirds in 3/4. However, we could have also written the answer as 2 remainder 1/12. This means that there are two 1/3s in 3/4, with a remainder of 1/12 (of 1). On the other hand, to write the answer as 2 1/12 is incorrect, because in this case we are mixing units — the 2 is 2 one-thirds, while the 1/12 is 1/12 of 1.

In summary, when using the measurement model of division, we are asking how many of the divisor (the second fraction) are in the dividend (the first fraction). The quotient tells us the number of copies of the divisor that are in the dividend.

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Copyright © 2006 by Daniel Siebert. All rights reserved. Students enrolled in MthEd 117 may make one (1) copy of this text for their personal use in this class. All other reproductions are expressly forbidden without the written permission of the author. For permission to use these materials as part of a class, please contact the author at dsiebert@mathed.byu.edu.