Lesson 8

Reveal one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.

The divisor for the second problem is half as much as the divisor in the first problem. Some students may think that the quotient will be half as much as well, instead of double. Remind the students of the two meanings of division they have learned and encourage them to draw a diagram to represent the problem.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have 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?”

This activity allows students to practice using the algorithm from earlier to solve division problems that involve a wider variety of fractions. Students can use any method of reasoning and are not expected to use the algorithm. As they encounter problems with less-friendly numbers, however, they notice that it becomes more challenging to use diagrams or other concrete strategies, and more efficient to use the algorithm. As they work through the activity, students choose their method.

Monitor the strategies students use and identify those with different strategies— including those who may not have used the algorithm—so they can share later.

Keep students in groups of 2. Give students 5–7 minutes of quiet work time, followed by 2–3 minutes to discuss their responses with a partner.

Select previously identified students to share their responses. Sequence their presentations so that students with the more concrete strategies (e.g., drawing pictures) share before those with more abstract strategies. Students using the algorithm should share last. Find opportunities to connect the different methods. For example, point out where the multiplication by a denominator and division by a numerator are visible in a tape diagram.

The purpose of this activity is to review different strategies for working with ratios and to prepare students to use these strategies with ratios involving fractions. The activity also foreshadows percentages by asking about the distance traveled in 100 minutes.

Monitor for different strategies like these:

• divide \(\frac{15}{2}\div 6\) to find the distance traveled in 1 minute, and then multiply it by 100.
• draw a double number line.
• create a table of equivalent ratios.

Depending on their prior learning, students might lean towards the first strategy.

Give students 3 minutes of quiet work time. Encourage them to find more than one strategy if they have time. Follow with whole-class discussion around the various strategies they used.

Students might calculate the unit rate as \(6\div \frac{15}{2}\). Ask students what this number would mean in this problem? (This number means that it takes \(\frac45\) of a minute to travel 1 kilometer.) In this case, students should be encouraged to create a table or a double number line, since it will help them make sense of the meaning of the numbers.

Select students to share the strategies they used. To the extent possible, there should be one student per strategy listed. If no students come up with one or more representations, create them so that students can compare and contrast.

• Divide (\(\frac{15}{2}\div 6\)) to find the number of kilometers traveled in 1 minute, and multiply by 100
• Double Number Line
• Table

Display strategies for all to see throughout the discussion.

Help students connect the strategies by asking:

• Was there a place in your solution where you calculated \(\frac{15}{2}\div 6\)?
• How can we see this value being used in the double number line? Table?

The purpose of this activity is to provide another context that leads students to calculate a unit rate from a ratio of fractions. This work is based on students’ work in grade 6 on dividing fractions.

Students notice and wonder about two statements and use what they wonder to create questions that are collected for all to see. Each student picks a question secretly and calculates the answer, then shares the answer with their partner. The partner tries to guess the question. Most of the time in this activity should be spent on students engaging in partner discussion.

Arrange students into groups of 2. Display the two statements for all to see. Ask students to write down what they notice and wonder, and then use what they wonder to come up with questions that can be answered using the given information. Create a list of questions and display for all to see. Here are suggested questions to listen for:

• Who ran faster, Noah or Lin?
• How far would Lin run in 1 hour?
• How far did Noah run in 1 hour?
• How long would it take Lin to run 1 mile at that rate?
• How long would it take Noah to run 1 mile at that rate?

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.

Students might have a hard time guessing their partner's question given only the answer.  Ask their partners to share the process they used to calculate the solution, they might leave out numbers and describe in general the steps they took to find the answer first. If their partner is still unable to guess the question, have them share the specific number they used. If they need additional support to guess the question, have their partner show them their work on paper (without sharing the question they answered) and see if this helps them figure out the question.

After both partners have a chance to guess each other’s question, ask a few different students to share their strategies for guessing which question their partner answered.