The goal of this activity is for students to discover that the balance point is another way to think about the mean, and that there is a relationship between measures of center and the shape of the data distribution.

Display the data set from the Launch again, and ask, “How could we find the actual balance point for this distribution without a meter stick and pennies?” (Find the mean of all the values represented in the dot plot.)

Here are sample questions to promote class discussion:

• “How are the balance points in the skewed examples different from the symmetric example?” (The balance point of the symmetric example is in the center, but the balance point of the skewed examples are not.)
• “What measure does the balance point represent?” (the mean)
• “What conclusions can you draw about measures of center and data distribution shapes?” (When the data distribution is symmetric, the mean and median will be equal and in the center of the distribution. When the distribution is skewed with a long tail off to one side, the mean may not be equal to the median.)

The purpose of this activity is for students to consider the mathematical language they are using to reason about which measure of center is more representative of a data set. 

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to which measure of center is a better representation of improvement. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback: 

• “ makes sense, but what do you mean when you say ?”

• “Can you describe that another way?”

• “Can you add numbers to support your reasoning?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, discuss how students decide which measure of center is the better representation. Here are sample questions to promote class discussion:

• “How do you decide which measure is a better representation of the data set?” (I think about which number summarizes the data set better.)
• “If you were on the marketing team for this gym, which measure of center would you use? What might you write on an advertisement stating this fact?” (The mean makes it sound like people improved more than the median, so I would use the mean. I might write, "An average member of our gym improved their score on a fitness test by 10 points in just 2 months!")
• “What are some patterns you noticed throughout the lesson between the distribution shapes and measures of center?” (The symmetric data tend to have the mean and median equal near what is typical for the data set, while skewed data often has a mean that is not.)
• “How does this skill of determining which measure of center is a better representation help you analyze a data set?” (This skill allows me to better analyze a data set because I have to think about how the data set is affected by extremes and how the shape of the data may change the measures of center.)