Lesson 4

The mathematical purpose of this warm-up is to collect informal terminology students may use to describe shapes of distributions, as well as any ways to describe distributions they may remember from work in earlier grades. This warm-up prompts students to compare four distributions. It gives students a reason to begin using language precisely (MP6) and gives you the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. Listen for students who use the statistically correct vocabulary as well as those who use informal language to describe the shapes.

Arrange students in groups of 2–4. Display the distributions 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, tell each student to share their reasoning why a particular item does not belong and together find at least one reason each item doesn't belong.

Select the identified groups to share their reason why a particular item does not belong so that those with informal language speak first and those with more precise terminology follow up. Ensure that each group shares 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 of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, recast any informal language that is used to describe the shape of each distribution. Introduce and define the terms symmetric, skewed, uniform, bimodal, and bell-shaped. It is important to note that the bell-shaped distribution is also symmetric.

The mathematical purpose of this activity is to give students a chance to practice finding data displays that represent the distribution of the same data set and using precise vocabulary for describing the shape of the distributions while taking turns matching cards. 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. Display the images of the dot plot and histogram. Ask students what they notice and wonder.

If it does not come up, help students notice that these two data displays show the same data in different formats. They can also be described as skewed right.

Give each group a set of cut-up cards. Explain that a match is two different displays that represent the distribution of the same set of data. Ask students to take turns: the first partner identifies a match, explains why they think it is a match, then describes the distribution while the other student listens and works to understand. Then they switch roles.

For students having trouble with the uniform distribution histograms, remind them that the lower bound for each interval is included and the upper bound is not. Ask them why this might change the last bar in each of these histograms. Some students may not know where to start to match data displays. You can tell them to look at the lowest and highest values as a starting point to finding similarities between two representations. 

Once all groups have completed the matching, discuss the following:

• “Which matches were tricky? Explain why.” (The uniform distributions may be difficult.)
• “What vocabulary was useful to describe the shape of the distribution?” (symmetric, skewed, uniform, bimodal, bell-shaped)
• “Were there any matches that could be described by more than one of these vocabulary terms?” (Yes, symmetric or skewed can also be used with some of the other terms for some of the distributions.)

If necessary, ask students to revoice less formal descriptions of the shape of the distribution using formal language including:

• Symmetric distribution
• Skewed distribution
• Uniform distribution
• Bimodal distribution
• Bell-shaped distribution

The mathematical purpose of this activity is to remind students of the importance of context to statistics. Although some analysis can be done outside of a context, it is often useful to think about the real situations in which the data was collected to engage student intuition and understanding.

Keep students in the same groups. Assign each pair of students one of the completed matches from the card sort activity. Tell students there are many possible answers for each representation. After 2 minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both reasonable. You may need to demonstrate this activity before beginning if you think students may have trouble getting started. After each group finishes with their assigned distribution, assign the group another distribution to consider.

Ask each group to share their response for at least one of the distributions they were assigned. After each group shares, ask the class if their context is reasonable. Here are some questions for discussion:

• “How did you use the shape of the data to come up with your question?” (Since the data was bell-shaped, I tried to think of situations where most of the data would be similar with a few points a little away from the these values.)
• “Would you always expect your question to result in a [symmetric, skewed, bell-shaped, etc.] distribution?” (Not necessarily, but for most cases it would.)

Reveal the actual survey question that produced the distribution. Actual questions by row:

1. How many points did Kiran score in each of his 22 games this season?
2. What were typical low temperatures in a Siberian town during January?
3. On a scale of 1–8, how was the service at the restaurant?
4. How many questions did people get correct on the vocabulary test the first week of school?
5. How many questions did people get correct on the vocabulary test the second week of school?
6. How many feet below the surface were each of the core samples taken?
7. How many trees are in my backyard at various temperatures?
8. What was the sum when you spun a spinner labeled 0 to 5 twice?
9. What was the weight of the crystal you grew in chemistry class?
10. How many questions did students get correct on a 10-item matching test?

Ask students to share what they have learned about the distribution now that they can think of the data in a real situation.