# Lesson 4

The Mean

## 4.1: Which One Doesn’t Belong: Division (5 minutes)

### Warm-up

This warm-up encourages students to analyze the structure and value of expressions, and to connect them to the process of calculating a mean.  Each expression has one obvious reason it does not belong, however, there is not one single correct answer.

As students discuss in small groups, listen for ideas related to finding the mean of a data set. Highlight these ideas during whole-class discussion.

### Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed one expression that does not belong and can explain why. When the minute is up, ask them to share their thinking with their small group, and then, together, find at least one reason each expression doesn't belong.

### Student Facing

Which expression does not belong? Be prepared to explain your reasoning.

$$\displaystyle \frac {8+8+4+4}{4}$$

$$\displaystyle \frac {10+10+4}{4}$$

$$\displaystyle \frac {9+9+5+5}{4}$$

$$\displaystyle \frac {6+6+6+6+6}{5}$$

### Activity Synthesis

Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question, attend to students’ explanations and ensure the reasons given are reasonable. If students give unsubstantiated claims, ask them to substantiate them.

At the end of the discussion, ask students which expression or expressions most represent how they would find the mean of a data set. There is a reason each expression (other than C) could represent how they would find the mean of a data set, however, highlight reasoning about the number of terms in the numerator being the same as the value of the denominator (e.g., there are 5 terms in the numerator and the denominator is 5).

## 4.2: Spread Out and Share (15 minutes)

### Activity

This activity introduces students to the concept of mean or average in terms of equal distribution or fair share. The two contexts chosen are simple and accessible, and include both discrete and continuous values. Diagrams are used to help students visualize the distribution of values into equal amounts.

The first set of problems (about cats in crates) can be made even more concrete by providing students with blocks or snap cubes that they can physically distribute into piles or containers.  Students using the digital activities will engage with an applet that allows students to sort cats. For the second set of problems (about hours of work), students are prompted to draw two representations of the number of hours of work before and after they are redistributed, creating a visual representation of fair shares or quantities being leveled out.

As students work, identify those with very different ways of arranging cats into crates to obtain a mean of 6 cats. Also look for students who determine the redistributed work hours differently. For example, some students may do so by moving the number of hours bit by bit, from a server with the most hours to the one with the fewest hours, and continue to adjust until all servers have the same number. Others may add all the hours and divide the sum by the number of servers.

### Launch

Arrange students in groups of 2. Provide access to straightedges. Also consider providing snap cubes for students who might want to use them to physically show redistribution of data values. If using the digital lesson, students will have access to an applet that will allow them to sort cats, snap cubes may not be necessary but can be provided.

Give students 3–4 minutes of quiet work time to complete the first set of questions and 1–2 minutes to share their responses with a partner. Since there are many possible correct responses to the question about the crates in a second room, consider asking students to convince their partner that the distribution that they came up with indeed has an average of 3 kittens per crate. Then, give students 4–5 min to work on the second set of questions together.

Representation: Develop Language and Symbols. Ensure access to virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with blocks or snap cubes that they can physically distribute into piles or containers.
Supports accessibility for: Conceptual processing

### Student Facing

1. The kittens in a room at an animal shelter are arranged in five crates, as shown. 1. The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?

2. The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate.

Explain how the expression $$10 \div 5$$ is related to the average.

3. Another room in the shelter has 6 crates. No two crates contain the same number of kittens, and there is an average of 3 kittens per crate.

Draw or describe at least two different arrangements of kittens that match this description. You may choose to use the applet to help.

2. Five servers were scheduled to work the number of hours shown. They decided to share the workload, so each one would work equal hours.
• Server A: 3
• Server B: 6
• Server C: 11
• Server D: 7
• Server E: 4
1. On the grid on the left, draw 5 bars whose heights represent the hours worked by servers A, B, C, D, and E. 2. Think about how you would rearrange the hours so that each server gets a fair share. Then, on the grid on the right, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
3. Based on your second drawing, what is the average or mean number of hours that the servers will work?
4. Explain why we can also find the mean by finding the value of $$31 \div 5$$.
5. Which server will see the biggest change to work hours? Which server will see the least change?

### Launch

Arrange students in groups of 2. Provide access to straightedges. Also consider providing snap cubes for students who might want to use them to physically show redistribution of data values. If using the digital lesson, students will have access to an applet that will allow them to sort cats, snap cubes may not be necessary but can be provided.

Give students 3–4 minutes of quiet work time to complete the first set of questions and 1–2 minutes to share their responses with a partner. Since there are many possible correct responses to the question about the crates in a second room, consider asking students to convince their partner that the distribution that they came up with indeed has an average of 3 kittens per crate. Then, give students 4–5 min to work on the second set of questions together.

Representation: Develop Language and Symbols. Ensure access to virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide students with blocks or snap cubes that they can physically distribute into piles or containers.
Supports accessibility for: Conceptual processing

### Student Facing

1. The manager of the shelter wants the kittens distributed equally among the crates. How might that be done? How many kittens will end up in each crate?

2. The number of kittens in each crate after they are equally distributed is called the mean number of kittens per crate, or the average number of kittens per crate. Explain how the expression $$10 \div 5$$ is related to the average.
3. Another room in the shelter has 6 crates. No two crates has the same number of kittens, and there is an average of 3 kittens per crate. Draw or describe at least two different arrangements of kittens that match this description.

2. Five servers were scheduled to work the number of hours shown. They decided to share the workload, so each one would work equal hours.

server A: 3

server B: 6

server C: 11

server D: 7

server E: 4

1. On the grid on the left, draw 5 bars whose heights represent the hours worked by servers A, B, C, D, and E. 2. Think about how you would rearrange the hours so that each server gets a fair share. Then, on the grid on the right, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
3. Based on your second drawing, what is the average or mean number of hours that the servers will work?
4. Explain why we can also find the mean by finding the value of the expression $$31 \div 5$$.

5. Which server will see the biggest change to work hours? Which server will see the least change?

### Student Facing

#### Are you ready for more?

Server F, working 7 hours, offers to join the group of five servers, sharing their workload. If server F joins, will the mean number of hours worked increase or decrease? Explain how you know.

### Anticipated Misconceptions

In the first room, to get each crate to have the same number of cats, some students might add new cats, not realizing that to “distribute equally” means to rearrange and reallocate existing quantities, rather than adding new quantities. Clarify the meaning of the phrase for these students.

Some students may not recognize that the hours for the servers could be divided so as to not be whole numbers. For example, some may try to give 4 servers 6 hours and 1 server has 7 hours. In this case, the time spent working is still not really divided equally, so ask the student to think of dividing the hours among the servers more evenly if possible.

### Activity Synthesis

Invite several students with different arrangements of cats in the second room with 6 crates to share their solutions and how they know the mean number of cats for their solutions is 3. Make sure everyone understands that their arrangement is correct as long as it had a total of 18 kittens and 6 crates and no two crates have the same number of cats. Show that the correct arrangements could redistribute the 18 cats such that there are 3 cats per crate.

Then, select previously identified students to share how they found the redistributed work hours if the workers were to spread the workload equally. Start with students who reallocated the hours incrementally (from one server to another server) until the hours level out, and then those who added the work hours and dividing the sum by 5.

Students should see that the mean can be interpreted as what each member of the group would get if everything is distributed equally, without changing the sum of values.

Representing: MLR7 Compare and Connect. After the selected students share their solutions, invite students to discuss “What is the same and what is different?” about the approaches. Call students’ attention to the connections between approaches by asking “How is the approach ‘divide the sum of work hours by 5’ represented visually in the new bar graph?” This will help students make sense of and use different representations of the mean of a set of data.
Design Principle(s): Maximize meta-awareness; Support sense-making

## 4.3: Travel Times (Part 2) (15 minutes)

### Activity

This activity serves two key purposes: to reinforce the idea of the mean as a balance point and a measure of center of a distribution, and to introduce the idea that distances of data points from the mean can help us describe variability in data, which prepares students to think about mean absolute deviation in the next lesson. Students also practice calculating mean of a distribution and interpreting it in context.

Unlike in previous activities, students are given less scaffolding for finding both the mean and the sums of distances from the mean. As students work, notice those who may need additional prompts to perform these tasks. Also listen for students’ explanations on what a larger mean tells us in this context. Identify those who can clearly distinguish how the mean differs from deviations from the mean.

Teacher Notes for IM 6–8 Accelerated

The activity Travel Times (Part 1) is not included in IM 6–8 Math Accelerated.

During the launch, tell students that a measure of center for a data distribution is a number that can be thought of as the middle or typical value of the distribution.

### Launch

Keep students in groups of 2. Give students 5 minutes of quiet work time to complete the first set of questions and then 2–3 minutes to discuss their responses with their partner before working on the second set of questions together.

The term “variation” is used in student text for the first time. If needed, explain that it has a similar meaning as “variability” and refers to how different or alike the data values are.

Representing, Conversing: MLR5 Co-craft Questions. Before revealing the task, display the image of Diego's and Andres’s dot plots, and only the first sentence of the problem statement. Ask students to write a list of mathematical questions that could be asked about what they see. Invite students to share their questions with a partner before selecting 2–3 to share with the class. Listen for questions that use the terms ‘mean’, ‘spread’, or ‘center’ and highlight where these are represented in the dot point. This helps students use mathematical language related to representing distributions of data sets and to understand the context of this problem prior to be asked to reason about the different quantities in the situation.
Design Principle(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

1. Here are dot plots showing how long Diego’s trips to school took in minutes—which you studied earlier—and how long Andre’s trips to school took in minutes. The dot plots include the means for each data set, marked by triangles.

1. Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?
2. Which of the two data sets has larger sums of distances to the left and right of the mean? What do these sums tell us about the variation in Diego’s and Andre’s travel times?
2. Here is a dot plot showing lengths of Lin’s trips to school.

1. Calculate the mean of Lin’s travel times.
2. Complete the table with the distance between each point and the mean as well whether the point is to the left or right of the mean.
time in minutes distance from the mean left or right of the mean?
22
18
11
8
11
3. Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.
4. Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times? What about the variability in their travel times?

### Activity Synthesis

The two big ideas to emphasize during discussion are: what the means tell us in this context, and what the sums of distances to either side of each mean tell us about the travel times.

Select a couple of students to share their analyses of Diego and Andre’s travel times. After each student explains, briefly poll the class for agreement or disagreement. If one or more students disagree with an analysis, ask for their reasoning and alternative explanations.

Then, focus the conversation how Lin and Andre’s travel times compare. Display the dot plots of their travel times for all to see.

Discuss:

• “How do the data points in Lin’s dot plot compare to those in Andre’s?”
• “How do their means compare? How do their sums of distances from the mean compare?”
• “What do the sums of distances tell us about the travel times?”
• “If more than half of Lin’s data points are far from the mean of 14 minutes, is the mean still a good description of her typical travel time? Why or why not?”

Students should see that larger distances from the mean suggest greater variability in the travel times. Even though both students have the same average travel time (both 14 minutes), Lin’s travel times are much more varied than Andre’s. A couple of Lin’s travel times are a lot longer or shorter than 14 minutes. Overall, her data points are within 6–8 minutes of the mean. For Andre, all of his data points are within 3 minutes of the mean.

The last discussion question prepares students to think about a different way to measure the center of a distribution in upcoming lessons.

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we look at finding the mean or the average of a numerical data set. We also learn that the mean is used as a measure of center of a distribution which is a number that seems typical of a distribution.
• “How do we find the mean of a data set?” (Add all of the values and divide the sum by the number of values in the data set.)
• “How can we interpret the mean of the heights of students in a class?” (Students in the class are about the value of the mean in height.)
• “Why might it make sense for the mean to be a number that describes the center of a distribution?” (Because the distances from the mean to the data points on the left of the mean are equal to the distances from the mean to the data points on the right of the mean, the mean tends to be in the middle of the data.)

## Student Lesson Summary

### Student Facing

Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use to summarize a distribution.

We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.

For example, suppose there are 5 bottles which have the following amounts of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.

To find the mean, first we add up all of the values. We can think of as putting all of the water together: $$1+4+2+3+0=10$$.

To find the “fair share,” we divide the 10 liters equally into the 5 containers: $$10\div 5 = 2$$.

Suppose the quiz scores of a student are 70, 90, 86, and 94. We can find the mean (or average) score by finding the sum of the scores $$(70+90+86+94=340)$$ and dividing the sum by four $$(340 \div 4 = 85)$$. We can then say that the student scored, on average, 85 points on the quizzes.
In general, to find the mean of a data set with $$n$$ values, we add all of the values and divide the sum by $$n$$.