Lesson 7

The purpose of this warm-up is to encourage students to connect the ideas they learned earlier about statistical questions and types of data (categorical and numerical) to the work on describing distributions (center and spread).

There are many ways to interpret the questions and identify how each one is unique. For example, they could say that Question A is about retirement, Question B about jobs, etc. If students begin to depart from thinking in statistical terms, remind them to think about how we might go about answering the questions and what the answers might involve. 

Arrange students in groups of 2–4. Display the questions for all to see. Give students 1 minute of quiet think time and ask students to indicate when they have noticed one question that does not belong and can explain why. When the minute is up, give students 2 minutes to share with their group their reasoning on why a question doesn't belong, and then, together, find at least one reason each question doesn't belong. 

Ask each group to share one reason why a particular question 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 for why a question does not belong, attend to students’ explanations and ensure the reasons given are correct.

If students use terms that are essential in this unit (such as center, spread, statistical, non-statistical, numerical data, categorical data, etc.), ask them to explain their meanings in their own words; these are opportunities to reinforce their understanding of the terms and to note any misconceptions. If students give unsubstantiated claims, ask them to substantiate them.

In a previous lesson, students had a primer on histograms—how they are drawn, how they differ from dot plots, and what information they can tell us. In this activity, students practice drawing a histogram for a given data set and using it to answer statistical questions. To help students understand the lengths involved in the data set, students are asked to draw various lengths used to group the worms in the first histogram.

As students organize the data set and draw their histogram, notice any challenges or questions students come across. If a question is raised by multiple groups, consider discussing it with the whole class. Also pay attention to how students use the histogram to identify a “typical” length—some might describe it in terms of the size of the bins (e.g., “a typical length is between 20 and 40 mm”); others might choose a value within a bin or a boundary between bins. Invite them to share their reasonings later.

Arrange students in groups of 2. Provide access to centimeter rulers.

Consider giving students a brief overview of the context for the problems in the activity. Tell students that there are nearly 6,000 species of earthworms in the world.  Some earthworms help the environment, while others (generally not native to the region in which they are found) may harm the environment.  Earthworms that are native to a particular region of the world are often raised, by farmers, in terrariums (a container or bin similar to an aquarium but it contains soil and leaves). The terrarium-raised earthworms provide bait for people who fish, provide food for various wildlife, and decompose food waste into soil.  Food waste and water are added to the terrariums as food for raising and growing worms.  Soil produced by the worms as they eat the food waste is often used as fertilizer. 

Explain that the lengths of the worms in the bins provide information about the ages of the worms, which can be useful for the farmer. In this activity, students will organize the lengths of the earthworms in several terrariums or bins. 

Give students 8–10 minutes of quiet work time, and then 3–4 minutes to discuss their work and complete the activity with a partner. 

When determining frequencies of data values, students might lose track of their counting. Suggest that they use tally marks to keep track of the number of occurrences for each bin.

When drawing the histogram, students might mistakenly use bar graphs as a reference and leave spaces between the bars. Ask them to look at the bars in other histograms they have seen so far and to think about what the gaps might mean considering that the bars are built on a number line.

Ask one or two students to display their completed histograms for all to see and briefly describe the overall distribution. Then, select a few other previously identified students to share their responses and explanations for the last two questions.

Focus the discussion on how identifying center and spread using a histogram is different than doing so using a dot plot. Discuss:

• “In a histogram, are we able to see clusters of values in the distribution?”
• “Can we see the largest and smallest values? Can we tell the overall spread?”
• “How do we identify the center of a distribution?”

From the various estimates that students give for a typical earthworm length (and from earlier exercises), students should begin to see that identifying a typical value of a distribution is not a straightforward or precise process so far. Explain that in upcoming lessons they will explore how to describe a typical value and characterize a distribution more systematically.

Now that students have some experience drawing and interpreting histograms, they use histograms to compare distributions of two populations. In a previous activity, students compared the two dot plots of students in a keyboarding class—one for the typing speeds at the beginning of the course and the other showing the speeds at the end of the course. In this activity, they recognize that we can compare distributions displayed in histograms in a similar way—by studying shapes, centers, and spreads.

Arrange students in groups of 2. Give students 4–5 minutes of quiet work time and 1–2 minutes to share their responses with a partner.

Select a few students to share their descriptions about basketball players and baseball players. After each student shares, ask others if they agree with the descriptions and, if not, how they might revise or elaborate on them. In general, students should recognize that the distributions of the two groups of athletes are quite different and be able to describe how they are different.

Highlight the fact that students are using approximations of center and different adjectives to characterize a distribution or a typical height, and that, as a result, there are variations in our descriptions. In some situations, these variations might make it challenging to compare groups more precisely. We will study specific ways to measure center and spread in upcoming lessons.