Making Sense of Division with Fractions

Post date: Nov 21, 2011 12:08:44 AM

Division with fractions is probably one of the least understood computations that students learn in their math lives. Traditionally, students have been taught to divide fractions by using the “flip and multiply” algorithm. While this method does produce correct answers, I find that students do not understand why it works and that it does not lead to sense-making. Thus, I show my students an alternative way to divide fractions.

Division can be interpreted two different ways. The divisor, or second number in a division problem, can be interpreted to mean “number of groups,” or it can be interpreted to mean “size of a group.” For example, 8 ÷ 2 can be interpreted to mean “put 8 in 2 groups” (the partitive division interpretation) or “put 8 into groups of 2” (the quotative division interpretation). In either instance, the meaning of the answer is different. If the divisor tells you how many groups you are making, the quotient, or answer, tells you the size of the group. If the divisor tells you how big each group is, the quotient tells you how many groups you make.

For certain fraction division problems, it is much easier to use one of the two interpretations of division. For example, 6/7 ÷ 3 can be thought of as breaking 6/7 into 3 groups. Each group would have 2/7 in it, so the quotient is 2/7. For another example, 2/3 ÷ 2/9 can be thought of as making groups of 2/9. Since 2/3 = 6/9, there are 3 groups of 2/9 in 6/9, so the quotient is 3.

Now, obviously I have picked problems for which it is somewhat intuitive to reason out an answer. That will not always be the case. Consider a problem like 1/2 ÷ 3/7. There is no easy way to reason through this problem at face value. If we, however, use equivalent fractions to get a common denominator, the problem becomes a little less murky. We can convert the numbers so that the problem is 7/14 ÷ 6/14. Now the question is, How many groups of 6/14 are there in 7/14? The answer is 7 ÷ 6, or 7/6, or 1 1/6 (1 whole group and 1/6 of another group). What about 2/5 ÷ 2/3? This is equivalent to 6/15 ÷ 10/15. Think of this problem as, If 10/15 is the size of a group, how much of a group is 6/15? The answer is 6 ÷ 10, or 6/10, or 3/5 (3/5 of a group).

Notice that to get the answer to these examples, we only had to divide the numerator of the first number by the numerator of the second number. Getting a common denominator allows us to focus only on the numerators in our computation. Although this algorithm may take a little longer than the “flip and multiply” method, I believe that it is one that leads to a better understanding of the computation and helps build sense-making in students. Please encourage your students to use this method even if they have previously been taught the "flip and multiply" algorithm.