standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
In this activity, students continue to develop their understanding of what could be considered typical for a group as well as variability in a data set. Students compare distributions with the same mean but different spreads and interpret them in the context of a situation. The context given here (basketball score) prompts them to connect the mean to the notion of how “well” a player plays in general, and deviations from the mean to how “consistently” that player plays.
They encounter the idea of calculating the
average absolute deviation from the mean
as a way to describe variability in data.
Launch
Arrange students in groups of 3–4. Give groups 6–7 minutes to answer the questions and follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Begin with a small-group or whole-class demonstration and think aloud to remind students how calculate mean using data from a dot plot.  Keep the worked-out calculations on display for students to reference as they work.
Supports accessibility for: Memory; Conceptual processing
Listening, Speaking: MLR2 Collect and Display.
Use this routine to collect student ideas and language about playing “well” and “consistently” as they discuss the players’ data in the three dot plots. Record students’ language on a display for all to see. This will help students connect their reasoning about the visual representations of data to statistical terms such as mean, deviation from the mean, and variability in data.
Design Principle(s): Support sense-making
Student Facing
Here are the dot plots showing the number of baskets Elena, Jada, and Lin each made over 12 school days.
On each dot plot, mark the location of the mean with a triangle (
\(\Delta\)
). Then, contrast the dot plot distributions. Write 2–3 sentences to describe the shape and spread of each distribution.
[IMG]
Discuss the following questions with your group. Explain your reasoning.
Would you say that all three students play equally well?
Would you say that all three students play equally consistently?
If you could choose one player to be on your basketball team based on their records, who would you choose?
Student Response
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Activity Synthesis
The purpose of the discussion is to highlight that the center of the distribution is not always the only consideration when discussing data. The variability or spread can also influence how we understand the data.
There are many ways to answer the second set of questions. Invite students or groups who have different interpretations of “playing well” and “playing consistently” to share their thinking. Allow as many interpretations to be shared as time permits. Discuss:
“How might we use the given data to quantify ‘playing well’ and ‘playing consistently?’”
“Is there a way to describe variability and consistency in playing precisely and in an objective way (rather than using broad, verbal descriptions)?”
Explain that we can describe variability more formally and precisely—using a number to sum it up; we will look at how to do so in the next activity.


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