Lesson 10

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.

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.

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).

In this activity, students explore the idea of the mean as a measure of center of all the values in the data, using a dot plot to help them visualize this idea. Students determine the distance between each data point and the mean, and notice that the sum of distances to the left is equal to the sum of distances to the right. In this sense, the mean can been seen as “balancing” the sets of points with smaller values than it and those with larger values. They make use of the structure (MP7) to calculate the distance between each data point and another point that is not the mean to see that the sums on the two sides are not equal. The idea of the mean as a measure of center of a distribution is introduced in this context.

As students work and discuss, identify those who could articulate why the mean can be considered a balancing point of a data set.

Arrange students in groups of 2. Give students 5 minutes to complete the first two questions with a partner, and then 5 minutes of quiet work time to complete the last two questions. Follow with a whole-class discussion.
 

Some students might write negative values for distances between the mean and points to the left of the mean. They might recall looking at distances between 0 and numbers to the left of it in a previous unit and mistakenly think that numbers to the left of the mean would have a negative distance from the mean. Remind students that distances are always positive; the answer to “How far away?” or “How many units away?” cannot be a negative number. 

Select a couple of students to share their observations on the distances between Diego's mean travel time and other points. To facilitate discussion, display this dot plot (with the distances labeled) for all to see. Discuss how the sums of distances change when different points are chosen as a reference from which deviations are measured.

Ask:

• “How are the sums of distances to mean (11 minutes) and the sums of distances to another point other than the mean (e.g., 10 minutes) different?”
• “If you choose another point or location on the number line, would it produce equal sums of distances to the left and to the right?”

Highlight the idea that only the mean could produce an equal sum of distances. Remind students that they have previously described centers of data sets. Explain that the mean is used as a measure of center of a distribution because it balances the values in a data set. Because data points that are greater than the mean balance with those that are less than the mean, the mean is used to describe what is typical for a data set.

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.  

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.

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.