Lesson 7

Finding an Algorithm for Dividing Fractions

Lesson Narrative

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).

Learning Goals

Teacher Facing

  • Generalize a process for dividing a number by a fraction, and justify (orally) why this can be abstracted as $n \boldcdot \frac{b}{a}$.
  • Interpret and critique explanations (in spoken and written language, as well as in other representations) of how to divide by a fraction.

Student Facing

Let’s look for patterns when we divide by a fraction.

Required Materials

Learning Targets

Student Facing

  • I can describe and apply a rule to divide numbers by any fraction.
  • I can divide a number by a non-unit fraction $\frac ab$ by reasoning with the numerator and denominator, which are whole numbers.

CCSS Standards

Building On


Glossary Entries

  • reciprocal

    Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).

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