Lesson 12

This number talk helps students to review strategies for dividing a decimal number by a whole number and to build their fluency. It also prepares them to calculate mean and MAD more efficiently. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

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 task. Follow with a whole-class discussion.

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?”

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.