This is the final activity in a series that leads students toward a general procedure for dividing fractions. Students verify previous observations about the steps for dividing non-unit fractions (namely, multiplying by the denominator and dividing by the numerator) and contrast the results with those found using tape diagrams. They then generalize these steps as an algorithm and apply it to answer other division questions.

As students discuss in their groups, listen to their observations and explanations. Select students with clear explanations to share later.

In the digital version of the activity, students can choose to use an applet to represent division of two fractions. The applet allows students to represent the fractions on two tape diagrams and measure how many times one fraction would fit in another. Students who need support in processing abstract visual information may benefit from the ability to represent the two fractions separately and partition each tape dynamically. If students don't have individual access, displaying the applet for all to see would be helpful.

Invite a couple of students to share their conclusion about how to divide a number by any fraction. Then, review the sequence of reasoning that led us to this conclusion using both numerical examples and algebraic statements throughout. Remind students that in the past few activities, we learned that:

Tell students that pairs of numbers such as 5 and ​​​​​,  and 7, and ​​​​ and ​​​​ are called reciprocals, or numbers that when multiplied equal 1. 

We can also say that 5 is a reciprocal of and is a reciprocal of 5. Ask students to name the reciprocal of a few other examples, such as 9, , , and

If time permits, ask students to use the generalized method to divide other fractions, such as or .

This activity serves two goals. It gives students opportunities to use the algorithm developed earlier to divide a wider variety of fractions. It also prompts them to consider the efficiency of different ways of finding quotients.

Students can use any method of reasoning and are not expected to use the algorithm. The numbers in the expressions are selected such that certain methods may be more practical than others. As students work through different problems, they notice that sometimes it is less convenient to use diagrams or other concrete strategies and more efficient to use the algorithm. Other times, it is easier to divide without using the algorithm. Students can choose an approach strategically by looking for and making use of structure in the dividends and divisors (MP7).

Monitor for students who use different ways to find quotients. Here are some likely approaches:

• Reasoning visually about equal groups, using a diagram.
• Reasoning numerically about equal groups, without using a diagram, such as by thinking “How many groups of this fraction is in that fraction?”
• Reasoning in terms of multiplication, such as by writing a corresponding multiplication equation for the division expression, or by thinking, “What number times the divisor gives the dividend?”
• Applying a generalization from an earlier course, such as by rewriting division by a whole number as multiplication by the reciprocal of that number.
• Applying the fraction division algorithm.

The purpose of this discussion is to highlight that, depending on the numbers involved, some ways to reason about fraction division may be more practical than others. Sometimes, such as when the denominator of the divisor is the same as or is a factor of that in the dividend, reasoning about equal groups—with or without diagrams—may be convenient. Other times, the fractions don’t lend themselves to be easily represented with diagrams or seen in terms of equal groups. Using a generalization or an algorithm might be much more productive.

Choose 3–4 problems that can illustrate different methods being favored for different problems. For instance, students may be more likely to opt for the algorithm to calculate the last two quotients than to find .

Invite previously selected students to share their responses and reasoning. Sequence the discussion of the approaches for each quotient from less common to more common. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “For which division problems was it quick to draw and use diagrams? For which problems were diagrams not convenient?”
• “For which division problems was the algorithm handy and preferable? For which problems was it less so?”
• “Were there divisions that you were able to do mentally without drawing a diagram or recording calculations on paper? What do you notice about the fractions in those problems?”