Lesson 6

The purpose of this warm-up is to connect the analytical work students have done with dot plots in previous lessons with statistical questions. This activity reminds students that we gather, display, and analyze data in order to answer statistical questions. This work will be helpful as students contrast dot plots and histograms in subsequent activities.

Arrange students in groups of 2. Give students 1 minute of quiet work time, followed by 2 minutes to share their responses with a partner. Ask students to decide, during partner discussion, if each question proposed by their partner is a statistical question that can be answered using the dot plot. Follow with a whole-class discussion. 

If students have trouble getting started, consider giving a sample question that can be answered using the data on the dot plot (e.g., “How many dogs weigh more than 100 pounds?”)

Ask students to share questions they agreed were statistical questions that could be answered using the dot plot. Record and display their responses for all to see. If there is time, consider asking students how they would find the answer to some of the statistical questions. 

Ask students to share a typical weight for a dog at this dog show and why they think it is typical. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree.

This activity introduces students to histograms. By now, students have developed a good sense of dot plots as a tool for representing distributions. They use this understanding to make sense of a different form of data representation. The data set shown on the first histogram is the same one from the preceding warm-up, so students are familiar with its distribution. This allows them to focus on making sense of the features of the new representation and comparing them to the corresponding dot plot.

Note that in all histograms in this unit, the left-end boundary of each bin or bar is included and the right-end boundary is excluded. For example, the number 5 would not be included in the 0–5 bin, but would be included in the 5–10 bin.

Explain to students that they will now explore histograms, another way to represent numerical data. Give students 3–4 minutes of quiet work time, and then 2–3 minutes to share their responses with a partner. Follow with a whole-class discussion.

Ask a few students to briefly share their responses to the first set of questions to make sure students are able to read and interpret the graph correctly.

Focus the whole-class discussion on the last question. Select a few students or groups to share their observations about whether or how their answers to the statistical questions would change if they were to use a dot plot to answer them, and about how histograms and dot plots compare. If not already mentioned by students, highlight that, in a histogram:

• Data values are grouped into “bins” and represented as vertical bars.
• The height of a bar reflects the combined frequency of the values in that bin.
• A histogram uses a number line.

At this point students do not yet need to see the merits or limits of each graphical display; this work will be done in upcoming lessons. Students should recognize, however, how the structures of the two displays are different (MP7) and start to see that the structural differences affect the insights we are able to glean from the displays. 

In this activity, students continue to develop their understanding of histograms. They begin to notice that a dot plot may not be best for representing a data set with a lot of variability (or where few values are repeated) or when a data set has a large number of values. Histograms may help us visualize a distribution more clearly in these situations. Students organize a data set into “bins” and draw a histogram to display the distribution. 

As students work and discuss, listen for explanations for why certain questions might be easy, hard, or impossible to answer using each graphical display. 

Give students a brief overview of census and population data, as some students may not be familiar with them. Refer to the dot plot of the population data and discuss questions such as:

• “How many total dots are there?” (51)
• “What's the population of the state with the largest population? Do you know what state that is?” (Between 37 and 38 million. It's California.)
• “Look at the leftmost dot. What state might it represent? Approximately what is its population?” (The leftmost dot represents Wyoming, with a population of around half a million.)
• “Do you know the approximate population of our state? Where do you think we are in the dot plot?”

Explain to students that they will now draw a histogram to represent the population data. Remind them that histograms organize data values into “bins” or groups. In this case, the bins sizes are already decided for them. Then, arrange students in groups of 3–4. Provide access to straightedges. Give students 10–12 minutes to complete the activity. Encourage them to discuss their work within their group as needed.

Much of the discussion about how to construct histograms should have happened in small groups. Address unresolved questions about drawing histograms if they are relatively simple. Otherwise, consider waiting until students have more opportunities to draw histograms in upcoming lessons.

Focus the discussion on comparing the effectiveness of dot plots and histograms to help us answer statistical questions.

Select a few students or groups to share how their ratings of “easy,” “hard,” and “impossible,” changed when they transitioned from using dot plots to using histograms to answer statistical questions about populations of states. Then, discuss and compare the two displays more generally. Solicit as many ideas and observations as possible to these questions:

• “What are some benefits of histograms?”
• “When might histograms be preferable to dot plots?”
• “When might dot plots be preferable to histograms?”

Students should walk away from the activity recognizing that in cases when a data set has many numerical values, especially if the values do not repeat, a histogram can give us a better visualization of the distribution. In such a case, creating a dot plot can be very difficult to do including finding a scale that can meaningfully display the data while a histogram will be easier to create and display the information in a way that is easier to understand at a glance.