Lesson 8
Ratios and Rates With Fractions
8.1: Number Talk: Dividing by Half as Much (5 minutes)
Warmup
Launch
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 wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Find each quotient mentally.
\(743 \div 10\)
\(743 \div 5\)
\(99 \div 3\)
\(99 \div \frac32\)
Student Response
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Anticipated Misconceptions
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.
If students get stuck on the last problem, help them see that previous problems can be used to figure out an answer to this last one. Since \(\frac32\) is half of 3 the answer is going to be double the previous answer.Activity Synthesis
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?”
Design Principle(s): Optimize output (for explanation)
8.2: Using an Algorithm to Divide Fractions (15 minutes)
Activity
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 lessfriendly 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.
Launch
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.
Supports accessibility for: Visualspatial processing; Organization
Student Facing
Calculate each quotient. Show your thinking and be prepared to explain your reasoning.
 \(\frac 89 \div 4\)
 \(\frac 34 \div \frac 12\)
 \(3 \frac13 \div \frac29\)
 \(\frac92 \div \frac 38\)
 \(6 \frac 25 \div 3\)
 After biking \(5 \frac 12\) miles, Jada has traveled \(\frac 23\) of the length of her trip. How long (in miles) is the entire length of her trip? Write an equation to represent the situation, and then find the answer.
Student Response
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Student Facing
Are you ready for more?
Suppose you have a pint of grape juice and a pint of milk. You pour 1 tablespoon of the grape juice into the milk and mix it up. Then you pour 1 tablespoon of this mixture back into the grape juice. Which liquid is more contaminated?
Student Response
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Activity Synthesis
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.
Design Principle(s): Maximize metaawareness; Optimize output (for comparison)
8.3: A Train is Traveling at . . . (10 minutes)
Activity
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.
Launch
Give students 3 minutes of quiet work time. Encourage them to find more than one strategy if they have time. Follow with wholeclass discussion around the various strategies they used.
Student Facing
A train is traveling at a constant speed and goes 7.5 kilometers in 6 minutes. At that rate:
 How far does the train go in 1 minute?
 How far does the train go in 100 minutes?
Student Response
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Anticipated Misconceptions
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.
Activity Synthesis
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?
Supports accessibility for: Language; Socialemotional skills
Design Principle(s): Maximize metaawareness
8.4: Comparing Running Speeds (10 minutes)
Activity
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.
Launch
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?
Supports accessibility for: Organization; Attention
Design Principle(s): Support sensemaking, Optimize output (for explanation)
Student Facing
Lin ran \(2 \frac34\) miles in \(\frac25\) of an hour. Noah ran \(8 \frac23\) miles in \(\frac43\) of an hour.
 Pick one of the questions that was displayed, but don’t tell anyone which question you picked. Find the answer to the question.
 When you and your partner are both done, share the answer you got (do not share the question) and ask your partner to guess which question you answered. If your partner can’t guess, explain the process you used to answer the question.
 Switch with your partner and take a turn guessing the question that your partner answered.
Student Response
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Student Facing
Are you ready for more?
Nothing can go faster than the speed of light, which is 299,792,458 meters per second. Which of these are possible?

Traveling a billion meters in 5 seconds.

Traveling a meter in 2.5 nanoseconds. (A nanosecond is a billionth of a second.)
 Traveling a parsec in a year. (A parsec is about 3.26 light years and a light year is the distance light can travel in a year.)
Student Response
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Anticipated Misconceptions
The warmup 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.
Activity Synthesis
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.
Lesson Synthesis
Lesson Synthesis
In this lesson, we worked with ratios of fractions.
 “What are strategies we can use to find solutions to ratio problems that involve fractions?” (double number line, tables, calculating unit rate)
 “How are those strategies different from and similar to ways we previously solved ratio problems that didn't involve fractions?” (They are structurally the same, but the arithmetic might take more time.)
8.5: Cooldown  Comparing Orange Juice Recipes (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
There are 12 inches in a foot, so we can say that for every 1 foot, there are 12 inches, or the ratio of feet to inches is \(1:12\). We can find the unit rates by dividing the numbers in the ratio:
\(1\div 12 = \frac{1}{12}\)
so there is \(\frac{1}{12}\) foot per inch.
\(12 \div 1 = 12\)
so there are 12 inches per foot.
The numbers in a ratio can be fractions, and we calculate the unit rates the same way: by dividing the numbers in the ratio. For example, if someone runs \(\frac34\) mile in \(\frac{11}{2}\) minutes, the ratio of minutes to miles is \(\frac{11}{2}:\frac34\).
\( \frac{11}{2} \div \frac34 = \frac{22}{3}\), so the person’s
pace is \(\frac{22}{3}\) minutes per mile.
\( \frac34 \div \frac{11}{2} = \frac{3}{22}\), so the person’s
speed is \(\frac{3}{22}\) miles per minute.