This Math Talk focuses on dividing by a fraction. It encourages students to think about the meaning of division and to rely on properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate unit rates for quantities with fractional values.

To apply reasoning from previous expressions to help evaluate the next expression, students need to look for and make use of structure (MP7).

To involve more students in the conversation, 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?”

The key takeaway is for students to have several methods for explaining why it makes sense that dividing by a fraction gives the same answer as multiplying by its reciprocal.

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.

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.)