Lesson 7

In this lesson students pull together the threads of reasoning from the previous four lessons to develop a general algorithm for dividing fractions. Students use tape diagrams to look at the effects of dividing by non-unit fractions. Through repetition, they notice a pattern in the steps of their reasoning (MP8) and structure in the visual representation of these steps (MP7). Students see that division by a non-unit fraction can be thought of as having two steps: dividing by the unit fraction, and then dividing the result by the numerator of the fraction. In other words, to divide by \(\frac25\) is equivalent to dividing by \(\frac15\), and then again by 2. Because dividing by a unit fraction \(\frac15\) is equivalent to multiplying by 5, we can evaluate division by \(\frac25\) by multiplying by 5 and dividing by 2.

Students calculate quotients using the steps they observed previously (i.e., to divide by \(\frac ab\), we can multiply by \(b\) and divide by \(a\)), and compare them to quotients found by reasoning with a tape diagram. Through repeated reasoning, they notice that the two methods produce the same quotient and that the steps can be summed up as an algorithm: to divide by \(\frac ab\), we multiply by \(\frac ba\) (MP8).