Lesson 18

This warm-up allows students to review two important ideas of this unit: interpreting data in a box plot and writing statistical questions based on a data set. Students write statistical questions based on given box plots, then trade questions to answer questions written by another student.

Arrange students in groups of 2. Tell students that, for the first question, one partner should write two questions about the head lengths and the other partner should write two questions about the head widths.  For the second question, they should exchange and review each other's questions. If their partner's question does not seem to be a statistical question, suggest a revision so that it becomes a statistical question, and then answer the question. Remind students to consider units of measurement.

Give students 2 minutes of quiet work time for the first question and 2 minutes for collaboration afterwards. 

Ask several students to share their questions about the head width and head length. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree it is a statistical question. If they agree, ask how they would find the answer, or for the answer itself. If they disagree, ask how they could rewrite the question so it is a statistical question. 

In this activity, students compare and contrast different measures of center and variability for data sets that have gaps and are not symmetrical. They interpret mean, MAD, median, and IQR in the context of a situation. Unlike many of the data sets students have seen so far, this one shows values that could roughly divide into three parts: the days when there is little or no homework, the days when there is a moderate number of homework problems, and the days when the assignment is relatively large. Because of this distribution, finding a typical number of homework problems (or whether it would be helpful to identify a typical number) is not obvious, prompting students to interpret measures of center and spread more carefully (MP2).

As students work and discuss, identify at least one student or group that decides that the mean and MAD are appropriate measures of center and spread and can explain their reasoning, and another that decides to go with the median and IQR and could support their choice. Invite them to share during whole-class discussion.

Briefly discuss students’ interpretations of the measures they just calculated:

• “What do the mean of 9 and MAD of 7.4 tell us? How can we interpret them in this context?”
• “What do the median of 7.5 and IQR of 14 tell us?”

Then, select two or more previously identified students to share their responses about which measures of center and spread are appropriate for summarizing the data set. After each person shares, briefly poll the class to see if others reasoned about the measures the same way. Sum up by asking:

• “Now that you have two pairs of measures of center and spread, how would you respond if someone asked you, ‘What is a typical number of homework problems for Mai’s class?’ Is the question easier to answer now?”

Students should walk away with increased awareness that, in some cases, measures of center and spread do not always paint a full picture of what the actual data set entails, and that the measures should be interpreted with care. 

In the previous activity, students considered appropriate measures of center and spread for describing distributions. Here, they show the same data set using three different kinds of graphical representations—a dot plot, a box plot, and histograms using different bin sizes—and decide which are more useful or more appropriate for communicating the distribution. 

As students work and discuss, identify those who draw clear graphical displays, those who noticed that the different displays offer different insights about the data distribution, and those who advocate for using different representations to display Jada's data. Ask them to share with the class later.

Invite previously identified students to share their dot plot, box plot, and histograms. Display their drawings for all to see. Then, select several students or groups to share their response to the last question (which representation should Jada choose?) and their explanation. If not already mentioned by students, discuss the different insights that each display offers, or different challenges it poses. (Some possible observations are listed under Student Response section.) For instance, consider asking the following questions about each data display:

• “What information can we get from this display?”
• “Does it give us a meaningful snapshot of the distribution?”
• “What characteristics of a different data set would make this representation more useful?”

Help students see that, in this case, none of the representations here are ill-suited to represent the data set, but a couple of them (e.g. the box plot, or the first histogram with a bin size of 10) allow us to describe the distribution the data set more easily because of how they summarize the data values in some ways.

In this culminating activity, students use what they have learned in the unit to answer statistical questions about a species of fish in the Great Lakes region. They use a histogram to represent the given data distribution, decide on appropriate measures of center and variability, and use their analyses to draw conclusions about a certain fish population.

Tell students that they will now look at an example in which data analysis could be used to help conservation efforts. Provide students with the following background information.

The yellow perch is a freshwater fish that is a popular food for people in the Great Lakes region (Minnesota, Wisconsin, Michigan, Illinois, Indiana, Ohio, Pennsylvania, and New York). In past research, samples of yellow perch taken from the Great Lakes seemed to be mostly male and mostly old. People worried that yellow perch might not survive and efforts were made to limit commercial and individual fishing in order to try to increase in the number of younger fish. An important part of these efforts is to periodically check the typical age of the fish in the Great Lakes.

The Wisconsin Department of Natural Resources and the Great Lakes Water Institute collected data from samples of yellow perch in Lake Michigan. Students at Rufus King High School in Milwaukee, Wisconsin participated in the research. They evaluated the data and presented their findings in a student-conducted press conference. Explain to students that, in this task, they will investigate some of the same questions that these students addressed in their research.  

Arrange students in groups of 3–4. Provide access to straightedges. Give students 7–8 minutes of quiet work time for the first three questions, and then 10–12 minutes to discuss their responses, complete the remainder of the task, and prepare a brief presentation on their response to the last set of questions.

Give each group access to tools for creating a visual display. Ask them to support their conclusions with specific pieces of evidence, such as their histogram, their analysis of the distribution, measures of center and spread, etc.

To allow all groups a chance to present, consider putting 2–3 groups together and asking them to present their work to each other. Groups that are not the first to present should focus on sharing new insights that have not been mentioned by the preceding groups. Invite students who are not presenting to attend carefully to the reasoning of the presenting group and to ask clarifying questions. 

If time permits, highlight some conclusions that students drew about whether the fish in the sample were young, adult age, or old, and whether researchers should be worried.

Tell students that several years after the students at Rufus King High School participated in the research, newer samples of yellow perch showed more favorable length-age distributions: more of the the fish were smaller or younger.