# 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.

### 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.

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 $$\frac52$$.