# Lesson 8

Ratios and Rates With Fractions

## 8.1: Number Talk: Dividing by Half as Much (5 minutes)

### Warm-up

The purpose of this number talk is to elicit strategies students have for reasoning about division. In the previous lesson, students learned an algorithm for dividing a fraction by a fraction. Later in this lesson, students will need to be able to divide a fraction by a fraction to solve problems about rates in contexts.

### 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 whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Find each quotient mentally.

$$743 \div 10$$

$$743 \div 5$$

$$99 \div 3$$

$$99 \div \frac32$$

### 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?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
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 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.

### 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.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to blank tape diagrams. Encourage students to attempt more than one strategy for at least one of the problems.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

Calculate each quotient. Show your thinking and be prepared to explain your reasoning.

1. $$\frac 89 \div 4$$
2. $$\frac 34 \div \frac 12$$
3. $$3 \frac13 \div \frac29$$
4. $$\frac92 \div \frac 38$$
5. $$6 \frac 25 \div 3$$
6. 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 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?

### 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.

Conversing, Representing: MLR7 Compare and Connect. As students consider the different strategies, invite them to make connections between the various representations and approaches. Ask, “What do each of the strategies have in common?”, “How are the strategies different?” and “Which strategy is more efficient? Why?” Listen for and amplify observations that include mathematical language and reasoning.
Design Principle(s): Maximize meta-awareness; 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 whole-class 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:

1. How far does the train go in 1 minute?
2. How far does the train go in 100 minutes?

### 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?
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because...”, “I noticed _____ so I...”, “Why did you...?”, “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR7 Compare and Connect. Use this routine when students present the strategies they used to determine the distance traveled. Ask students to consider what is the same and what is different about each approach. Draw students' attention to the different ways the unit rate and total distance can be seen in each representation (i.e., tape diagram, double number line and table). Listen for and amplify students' correct use of the term “unit rate.” These exchanges can strengthen students' mathematical language use as they reason to make sense of strategies used to calculate unit rates and distance traveled.
Design Principle(s): Maximize meta-awareness

## 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?
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking: MLR8 Discussion Supports. Provide sentence frames to students to support their explanations of their process for finding the answer to their selected question. For example, “First, I _____. Then, I ____.”
Design Principle(s): Support sense-making, 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.

1. Pick one of the questions that was displayed, but don’t tell anyone which question you picked. Find the answer to the question.
2. 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.

### 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?

1. Traveling a billion meters in 5 seconds.

2. Traveling a meter in 2.5 nanoseconds. (A nanosecond is a billionth of a second.)

3. 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.)

### Anticipated Misconceptions

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.

### 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.)

## 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.