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?”

In this activity students solve a problem involving constant speed. Both the distance and the time are given as fractions, giving students the opportunity to divide a fraction by a fraction to calculate the unit rate. As students calculate equivalent rates and compare the rates, they are reasoning quantitatively and abstractly (MP2).

Monitor for students who use these different strategies to compare the two runners’ speeds:

• Find the amount of time it would take each person to run the same distance.
• Find the distance each person could run in the same amount of time.
• Calculate a unit rate for each person. (This is a special case of one of the two previous bullets, depending on whether the unit rate represents the person’s speed or pace.)

The Warm-up was intended to remind students of some strategies for dividing fractions by fractions, but students may need additional support working with the numbers in this task.

Invite students to share how they determined whether Noah or Lin ran faster. Ask students to describe similarities and differences between the different strategies and representations used.

To involve more students in the discussion, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

Highlight any places in the solution process where students divided fractions. Ask students to describe how they found the quotient.