The Shape of Distributions
10 min
Narrative
This is the first Which Three Go Together routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together?" and "Why do they go together?”
Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepen their awareness of connections across representations.
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.
Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Launch
Arrange students in groups of 2–4. Display the dot plots for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three dot plots that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
- A, B, and C go together because they are symmetric (not skewed to one side).
- A, B, and D go together because they have different amounts of dots over different numbers (they are not uniform).
- A, C, and D go together because the median is a good description of typical values in the distribution.
- B, C, and D go together because the data are not clustered near the center.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three goes together. 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 three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any statistical terminology they use, such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “bell-shaped,” and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
Standards
- HSS-ID.A.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- HSS-ID.A.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
15 min
10 min
Knowledge Components
All skills for this lesson
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The Shape of Distributions
10 min
Narrative
This is the first Which Three Go Together routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together?" and "Why do they go together?”
Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepen their awareness of connections across representations.
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.
Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Launch
Arrange students in groups of 2–4. Display the dot plots for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three dot plots that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
- A, B, and C go together because they are symmetric (not skewed to one side).
- A, B, and D go together because they have different amounts of dots over different numbers (they are not uniform).
- A, C, and D go together because the median is a good description of typical values in the distribution.
- B, C, and D go together because the data are not clustered near the center.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three goes together. 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 three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any statistical terminology they use, such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “bell-shaped,” and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
Standards
- HSS-ID.A.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- HSS-ID.A.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.