# Lesson 6

Per Each

## 6.1: Number Talk: Dividing by Powers of 10 (10 minutes)

### Warm-up

This number talk encourages students to use the structure of base ten numbers to find the quotient of a base ten number and 10. The goal is to get students to see how understanding each quotient helps them find the next quotient. Reasoning about this computation will be important in both this lesson and future lessons where students are working with the metric system and percentages.

Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 5 minutes.

### Launch

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. 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 the quotient mentally.

$$30\div 10$$

$$34\div 10$$

$$3.4\div 10$$

$$34\div 100$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Emphasize student strategies based in place value to explain methods students may have learned about “moving the decimal point” left or right or “crossing out zeros.” To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone solve the problem the same way 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)

## 6.2: More Shopping (15 minutes)

### Activity

In this task, students practice finding unit prices, using different reasoning strategies, and articulating their reasoning. They also learn about the term “at this rate.”

As students work, observe their work and then assign one problem for each group to own and present to the class. (The problems can each be assigned to more than one group). Have them work together to create a visual display of their problem and its solution.

Teacher Notes for IM 6–8 Accelerated

Adjust this activity to 10 minutes.

In the launch, adjust the displayed problem to: “Pizza costs $10 for 8 slices. At this rate, how much will 6 slices cost?” During the discussion of the phrase “at this rate,” ask students to outline a method for solving the problem. Highlight any methods that determine the cost for 1 slice of pizza. Tell students that the price for 1 item is called the unit price and can be described using the word “per.” For example, “This pizza costs$1.25 per slice.”

Tell students to omit the last question in which they create a visual display.

### Launch

Arrange students in groups of 3–4. Provide tools for creating a visual display and access to rulers. Explain that they will work together to solve some shopping problems, run their work by you, and prepare to present an assigned problem to the class. Tell students that they can use double number lines if they wish.

Display the problem and read it aloud: Pizza costs $1.25 per slice. At this rate, how much will 6 slices cost? Ask students what they think “at this rate” means in the question. Ensure they understand that “at this rate” means we know that equivalent ratios are involved: • The ratio of cost to number of slices is$1.25 to 1. That is, pizza costs $1.25 per slice. • The ratio of cost to number of slices is something to 6. That is, pizza costs something for 6 slices. The something is the thing we are trying to figure out, and “at this rate” tells us that the two ratios in this situation are equivalent. Another way to understand “at this rate” in this context is “at this price per unit” and that the price per unit is the same no matter how many items or units are purchased. Discuss any expectations for the group presentation. For example, each group member might be assigned a specific role for the presentation. If students have digital access, they can use an applet to explore the problems and justify their reasoning before preparing their presentations. ### Student Facing 1. Four bags of chips cost$6.

1. What is the cost per bag?

2. At this rate, how much will 7 bags of chips cost?

2. At a used book sale, 5 books cost $15. 1. What is the cost per book? 2. At this rate, how many books can you buy for$21?

3. Neon bracelets cost $1 for 4. 1. What is the cost per bracelet? 2. At this rate, how much will 11 neon bracelets cost? Pause here so your teacher can review your work. 4. Your teacher will assign you one of the problems. Create a visual display that shows your solution to the problem. Be prepared to share your solution with the class. Here is an applet you may use if you choose to. ### Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. ### Launch Arrange students in groups of 3–4. Provide tools for creating a visual display and access to rulers. Explain that they will work together to solve some shopping problems, run their work by you, and prepare to present an assigned problem to the class. Tell students that they can use double number lines if they wish. Display the problem and read it aloud: Pizza costs$1.25 per slice. At this rate, how much will 6 slices cost?

Ask students what they think “at this rate” means in the question. Ensure they understand that “at this rate” means we know that equivalent ratios are involved:

• The ratio of cost to number of slices is $1.25 to 1. That is, pizza costs$1.25 per slice.
• The ratio of cost to number of slices is something to 6. That is, pizza costs something for 6 slices.

The something is the thing we are trying to figure out, and “at this rate” tells us that the two ratios in this situation are equivalent. Another way to understand “at this rate” in this context is “at this price per unit” and that the price per unit is the same no matter how many items or units are purchased.

Discuss any expectations for the group presentation. For example, each group member might be assigned a specific role for the presentation.

If students have digital access, they can use an applet to explore the problems and justify their reasoning before preparing their presentations.

### Student Facing

1. Four bags of chips cost $6. 1. What is the cost per bag? 2. At this rate, how much will 7 bags of chips cost? 2. At a used book sale, 5 books cost$15.

1. What is the cost per book?
2. At this rate, how many books can you buy for $21? 3. Neon bracelets cost$1 for 4.

1. What is the cost per bracelet?
2. At this rate, how much will 11 neon bracelets cost?

4. Your teacher will assign you one of the problems. Create a visual display that shows your solution to the problem. Be prepared to share your solution with the class.

### Anticipated Misconceptions

The first and third questions involve using decimals to represent cents. If the decimal point is forgotten, remind students that the cost of the bracelet is less than one dollar, and the cost of the chips is in between one and two dollars.

Watch for students working in cents instead of dollars for the bracelets. They may come up with an answer of 275 cents. For these students, writing 25 cents as \$0.25 should help, or consider reminding them of the avocados from a previous activity, which had a unit price of \$0.50.

### Activity Synthesis

Invite each group to present its assigned problem. After each group presents, highlight the group’s strategy, accurate uses of the terms “at this rate” and “per,” and the ways in which a double number line might have been used when working with unit price.

## 6.3: Moving 10 Meters (25 minutes)

### Activity

This activity gives students first-hand experience in relating ratios of time and distance to speed. Students time one another as they move 10 meters at a constant speed—first slowly and then quickly—and then reason about the distance traveled in 1 second.

Double number lines play a key role in helping students see how time and distance relate to constant speed, allowing us to compare how quickly two objects are moving in two ways. We can look at how long it takes to move 10 meters (a shorter time needed to move 10 meters means faster movement), or at how far one travels in 1 second (a longer distance in one second means faster movement).

Along the way, students see that the language of “per” and “at this rate,” which was previously used to talk about unit price, is also relevant in the context of constant speed. They begin to use “meters per second” to express measurements of speed.

As students work, notice the different ways they use double number lines or other means to reason about distance traveled in one second.

Teacher Notes for IM 6–8 Accelerated
Adjust the timing of this activity to 20 minutes.

### Launch

Before class, set up 4 paths with a 1-meter warm-up zone and a 10-meter measuring zone.

Arrange students into 4 groups, with one for each path. Provide a stopwatch. Explain that they will gather some data on the time it takes to move 10 meters. Select a student to be your partner and demonstrate the activity for the class.

• Share that the experiment involves timing how long it takes to move the distance from the start line to the finish line.
• Explain that each person in the pair will play two roles: “the mover” and “the timer.” Each mover will go twice—once slowly and once quickly—starting at the warm-up mark each time. The initial 1-meter-long stretch is there so the mover can accelerate to a constant speed before the timing begins.
• Demonstrate the timing protocol as shown in the task statement.

Stress the importance of the mover moving at a constant speed while being timed. The warm-up segment is intended to help them reach a steady speed. To encourage students to move slowly, consider asking them to move as if they are balancing something on their head or carrying a full cup of water, trying not to spill it.

Alternatively, set up one path and ask for two student volunteers to demonstrate while the rest of the class watches.

Representation: Internalize Comprehension. Begin with a physical demonstration of the activity. Highlight connections between prior understandings about using number lines to show how time and distance relate to constant speed.
Supports accessibility for: Conceptual processing; Visual-spatial processing

### Student Facing

Your teacher will set up a straight path with a 1-meter warm-up zone and a 10-meter measuring zone. Follow the following instructions to collect the data.

1. The person with the stopwatch (the “timer”) stands at the finish line. The person being timed (the “mover”) stands at the warm-up line.
2. On the first round, the mover starts moving at a slow, steady speed along the path. When the mover reaches the start line, they say, “Start!” and the timer starts the stopwatch.
3. The mover keeps moving steadily along the path. When they reach the finish line, the timer stops the stopwatch and records the time, rounded to the nearest second, in the table.
4. On the second round, the mover follows the same instructions, but this time, moving at a quick, steady speed. The timer records the time the same way.
5. Repeat these steps until each person in the group has gone twice: once at a slow, steady speed, and once at a quick, steady speed.
1. After you finish collecting the data, use the double number line diagrams to answer the questions. Use the times your partner collected while you were moving.

Moving slowly:

Moving quickly:

1. Estimate the distance in meters you traveled in 1 second when moving slowly.
2. Estimate the distance in meters you traveled in 1 second when moving quickly.
3. Trade diagrams with someone who is not your partner. How is the diagram representing someone moving slowly different from the diagram representing someone moving quickly?

### Anticipated Misconceptions

Students may have difficulty estimating the distance traveled in 1 second. Encourage them to mark the double number line to help. For example, marking 5 meters halfway between 0 and 10 and determining the elapsed time as half the recorded total may cue them to use division.

### Activity Synthesis

Select students to share who used different methods to reason about the distance traveled in 1 second. It may be helpful to discuss the appropriate amount of precision for their answers. Dividing the distance by the elapsed time can result in a quotient with many decimal places; however, the nature of this activity leads to reporting an approximate answer.

During the discussion, demonstrate the use of the phrase meters per second or emphasize it, if it comes up naturally in students’ explanations.  Discuss how we can use double number lines to distinguish faster movement from slower movement. If it hasn't already surfaced in discussion, help students see we can compare the time it takes to travel the same distance (in this case, 10 meters) as well as the distance traveled in the same amount of time (say, 1 second).

Explain to students that when we represent time and distance on a double number line, we are saying the object is traveling at a constant speed or a constant rate. This means that the ratios of meters traveled to seconds elapsed (or miles traveled to hours elapsed) are equivalent the entire time the object is traveling. The object does not move faster or slower at any time. The equal intervals on the double number line show this steady rate.

Representing: MLR7 Compare and Connect. As students share different approaches for reasoning about distance traveled in 1 second, ask students to identify "what is the same and what is different?" about the approaches. Help students connect approaches by asking "Where do you see the measurement of speed '____ meters per second' in each approach?" This helps students connect the concept of rate and a visual representation of that rate.
Design Principle(s): Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

The main ideas to develop in this lesson are techniques for finding a unit price and speed, and the things that can be done once the unit price or speed is known.

Discuss with students the methods they use to find a unit price. The likely answers are:

• Division: if 2 bags of rice cost $3, then 1 bag costs$1.50 because $$3 \div 2 = 1.5$$.
• Double number line: adding tick marks to a double number line signifying 1 bag can determine the cost per bag. Briefly discuss with students the meaning of the word per (for each).

Knowing a speed in meters per second gives the same kind of information as knowing a unit price in dollars per item. The overall objective is for students to see consistencies in the underlying mathematical structure of these contexts.

Discuss with students the things they can do once they know a unit price. Specifically, they can directly compute any cost (or distance traveled) when the number of items (or time spent moving) is known by multiplying the unit price (or constant speed) by the number of items (or time spent moving). You may want to point out to students that, by multiplying, they are finding part of an equivalent ratio. For example, the ratio “$30 for 20 bags“ is equivalent to the ratio “$3 for 2 bags.”

## Student Lesson Summary

### Student Facing

The unit price is the price of 1 thing—for example, the price of 1 ticket, 1 slice of pizza, or 1 kilogram of peaches.

If 4 movie tickets cost $28, then the unit price would be the cost per ticket. We can create a double number line to find the unit price. This double number line shows that the cost for 1 ticket is$7. We can also find the unit price by dividing, $$28 \div 4 = 7$$, or by multiplying, $$28 \boldcdot \frac{1}{4} = 7$$.

Double number lines can also be used to make sense of objects moving at a constant speed. Suppose a train traveled 100 meters in 5 seconds at a constant speed.

The double number line shows that the train’s speed was 20 meters per second. We can also find the speed by dividing: $$100 \div 5 = 20$$.
Once we know the speed in meters per second, many questions about the situation become simpler to answer because we can multiply the amount of time an object travels by the speed to get the distance. For example, at this rate, how far would the train go in 30 seconds? Because $$20 \boldcdot 30 = 600$$ , the train would go 600 meters in 30 seconds.