The purpose of this activity is for students to think about the mean of a data set as the balance point of a distribution. Students should arrive at the balance point by mentally thinking about the point where the distribution would balance. This activity prepares students to recognize the relationship between mean and median by looking at the shape of the distribution in a later lesson. Students reason abstractly and quantitatively (MP2) when they make a connection between balance point and mean.

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 practice making a distinction between which measure of center is more representative of a data set. For each problem, students answer questions about the mean and median and decide which number seems more representative of the given data. This will prepare students for a later lesson when they have to recognize the relationship between a data distribution shape and measures of center. Students reason abstractly and quantitatively (MP2) when they decide which measure of center more closely represents what is typical of a data set.

This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by _____?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.

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.)