Lesson 4

This warm-up prompts students to compare four distributions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Arrange students in groups of 2–4. Display the data displays for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as measure of center, measure of variability, bell-shaped, bimodal, or outliers. Also, press students on unsubstantiated claims.

This activity is optional. It reviews how to describe distributions based on words such as “bell-shaped” and “symmetric.” If students understand these concepts well, this activity may be safely skipped.

In this activity, students take turns with a partner describing distributions of data represented in dot plots. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Arrange students in groups of 2. Tell students that for each dot plot, one partner describes the distributions and explains their reasoning. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next dot plot, the students swap roles. If necessary, demonstrate this protocol before students start working.

The purpose of this discussion is for students to describe various distributions. Here are some questions for discussion.

• “How did you describe the fourth distribution?” (It was really hard to describe, but my partner and I agreed that it was close to uniform and close to being symmetric.)
• “Is approximately symmetric a good way to describe the fourth distribution? Explain your reasoning.” (Yes, it is a good way to describe it because then I know it may not be exactly symmetric, but is very close to being symmetric. The dot plot from the warm-up is also approximately symmetric.)
• “What does it mean to describe a distribution as approximately uniform?” (It means that instead of each value having exactly the same frequency, they all have frequencies that are almost exactly the same.)
• “Draw a dot plot that is approximately bell-shaped. Compare dot plots with a partner.”

This activity is optional. It reviews how to match mean, median, and standard deviation to a distribution. If students understand the roles of measures of center and standard deviation, this activity may be safely skipped.

In this activity, students take turns with a partner matching data represented by dot plots with summary statistics. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

Arrange students in groups of 2. Tell students that for each dot plot, one partner describes how they determined which summary statistics match the data in their dot plot. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next dot plot, the students swap roles. If necessary, demonstrate this protocol before students start working.

Students may not recall the meaning of standard deviation in this situation. Ask students to recall the definition of standard deviation as it is related to how the data are spread. Standard deviation will be more important later in this unit, so it will be important for them to remember its meaning as spread of the distribution.

The goal of this discussion is for students to understand the relationship between the shape of a distribution and the mean, median, and standard deviation. Here are some questions for discussion:

• “Dot plot A and B have the same mean and median. How did you figure out which one had a greater standard deviation?” (I figured out that the dot plot A had to have a greater standard deviation than dot plot B because more of the values were further from the mean of 5, whereas dot plot B had several values equal to 5 and fewer further from 5. Since the standard deviation is calculated using distances from the mean, it makes sense that dot plot A has a greater standard deviation than dot plot B.)
• “Dot plot C and D are skewed right and skewed left, respectively. What does this tell you about the relationship between the mean and the median for each graph?” (For dot plot C, I expected the median to be less than the mean because it would be less impacted by the values to the right in the dot plot, whereas the mean would be expected to increase because of those values. For dot plot D, I expected the mean to be less than the median because it is disproportionately impacted by the values to the left in the dot plot.)
• "What is the standard deviation measuring? Explain your reasoning.” (The standard deviation is measuring the variability relative to the mean.)