Lesson 5

The purpose of this warm-up is for students to first reason about the mean of a data set without calculating and then practice calculating mean. The context will be used in an upcoming activity in this lesson so this warm-up familiarizes students with the context for talking about deviation from the mean. 

In their predictions, students may think that Elena will have the highest mean, because she has a few very high scores (7, 8, and 9 points). They may also think that Lin and Jada will have very close means because they each have 5 higher scores than one another and their other scores are the same. Even though each player has the same mean, all of these ideas are reasonable things for students to consider when looking at the data. Record and display their predictions without further questions until they have calculated and compared the mean of their individual data sets. 

Arrange students in groups of 3. Display the data sets for all to see. Ask students to predict which player has the largest mean and which has the smallest mean. Give students 1 minute of quiet think time and then poll students on the player who they think has the largest and smallest mean. Ask a few students to share their reasoning.

Tell each group member to calculate the mean of the data set for one player in the task, share their work in the small group, and complete the remaining questions. 

Ask students to share the mean for each player's data set. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree and what the mean tells us in this context. If the idea that the means show that all three students make, on average, half of the 10 attempts to get the basketball in the hoop does not arise, make that idea explicit. 

If there is time, consider revisiting the predictions and asking how the mean of Elena's data set can be the same as the others when she more high scores?

In the previous activity, students evaluated the performance of three students based on the mean and variability. Here they learn the term mean absolute deviation (MAD)as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variations in context.

While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in grade 6. 

The activity Shooting Hoops (Part 2) and the activity referred to in the launch are not included in IM 6–8 Math Accelerated.

During the launch, ask students what they think will be true about distances between data points and the mean, knowing that the mean can be considered a balance point. (The sum of the distances to the left of the mean is the same as the sum of the distances to the right of the mean.) Then tell students that the distances can also tell us something else about a distribution that will be explored in this activity.

Remind students in the last lesson they found distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right were equal, which allows us to think of the mean as the balancing point or the center of the data. Explain that the distance between each point and the mean can tell us something else about a distribution. 

Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.

Students may recall the previous lesson about thinking of the mean as a balance point and think that the MAD should always be zero since the left and right distances should be equal. Remind them that distances are always positive, so finding the average of these distances to the mean can only be zero if all the data points are exactly at the mean.

During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:

• “What can we say about a data set whose data points have very small distances from the mean?”
• “What about a data set with points that show large distances from the mean?”
• “Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?”
• “What do MAD values of 2, 1.5, and 1 mean in this context?”

This activity allows students to practice calculating MAD and build a better understanding of what it tells us. Students continue to compare data sets with the same mean but different MADs and interpret what these differences imply in the context of the situation.

Arrange students in groups of 3–4. Before students read the task statements, display the two dot plots in the task for all to see. Give students up to 1 minute to study the dot plots and share with their group what they notice and wonder about the plots.

Next, select a few students to share their observations and questions; it is not necessary to confirm or correct students' observations or answer their questions at this point. If no one noticed or wondered about what variable the data sets show, ask students what they think the context for the data sets might be or what quantities they show. Explain to students that they will find more information in the task statement to help them compare and interpret the dot plots. 

Give students 3–4 minutes of quiet work time to complete the first set of questions, and then 8–10 minutes to complete the second set with their group. Allow at least a few minutes for a whole-class discussion.

Select a couple of students to share their responses to the first set of questions about how they matched the dot plots to the players (Andre and Noah) and how they knew.

Then, display a completed table and the MAD for the second set of questions. Give students a moment to check their work. To facilitate discussion, help students connect MAD and the spread of data, and enable them to make comparison, consider displaying all three dot plots at the same scale and using a line segment to represent the MAD on each dot plot, as shown here.

Invite a few students to share their observations about how the means and MADs of Noah and the eighth-grade student compare. Discuss:

• “How are the distributions of points related to the mean? How are they related to the MAD?”
• “Which might be more desirable for a basketball team: a lower mean or a higher mean? Why?”
• “Which might be more desirable: a lower MAD or a higher MAD? Why?”
• “Of the three students, which one would you want on your team? Why?”

Expect students to choose different players to be on their team, but be sure they support their preferences with a reasonable explanation (MP3). Students should walk away understanding that, in this context, a higher MAD indicates more variability and less consistency in the number of shots made. 

In this activity, students continue to practice interpreting the mean and the MAD and to use them to answer statistical questions. A new context is introduced, but students should continue to consider both the center and variability of the distribution as ways of thinking about what is typical for a set of data and how consistent the data tends to be.

Give students 5–7 minutes of quiet work time. Ask students to consider drawing a triangle and a line segment on each dot plot in the last question to represent the mean and MAD for each data set (as was done in an earlier lesson). 

Have a student display dot plots with the means and MADs marked on them. Invite several students to share their comments about how the composition of the swimming team has changed over the three decades. Some discussion questions:

• “What can you say about the size of the team? Has it changed?”
• “Overall, has the team gotten older, younger, or stayed about the same? How do you know?”
• “Has the age of the youngest swimmer changed? What about the age of the oldest swimmer?”
• “Has the team become more diverse in ages, in general? Or have the swimmers become more alike in their age? How do you know?”

Students should recognize that the higher mean and MAD for the 2016 swimming team tell us that the team, on the whole, has gotten older and more diverse in ages. In 1984, 18.2 years was a typical age for the swimmers. A typical age for the 2016 swimmers was 22.8 years and there was a wider range of ages represented.