# Lesson 3

Interpreting Histograms

## 3.1: Dog Show (Part 1) (5 minutes)

### Warm-up

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.

### Launch

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?”)

### Student Facing

Here is a dot plot showing the weights, in pounds, of 40 dogs at a dog show.

1. Write two statistical questions that can be answered using the dot plot.
2. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.

### Activity Synthesis

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.

## 3.2: Dog Show (Part 2) (10 minutes)

### Activity

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.

### Launch

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.

### Student Facing

Here is a histogram that shows some dog weights in pounds.

Each bar includes the left-end value but not the right-end value. For example, the first bar includes dogs that weigh 60 pounds and 68 pounds but not 80 pounds.

1. Use the histogram to answer the following questions.

1. How many dogs weigh at least 100 pounds?

2. How many dogs weigh exactly 70 pounds?

3. How many dogs weigh at least 120 and less than 160 pounds?

4. How much does the heaviest dog at the show weigh?

5. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
2. Discuss with a partner:

• How are the histogram and the dot plot alike? How are they different?

### Activity Synthesis

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.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary.  Include the following terms and maintain the display for reference throughout the unit: histogram.
Supports accessibility for: Memory; Language
Representing, Listening, Speaking: MLR2 Collect and Display. Record and display the language students use as they discuss how dot plots and histograms are alike and different. Highlight language related to individual specific data values and groups of data values, center, and spread. Keep the display visible for reference in upcoming lessons. This will help students connect unique and shared characteristics of histograms and dot plots.
Design Principle(s): Support sense-making; Maximize meta-awareness

## 3.3: Tall and Taller Players (10 minutes)

### Activity

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.

### Launch

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.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Histogram ___ represents the heights of _______ players because . . .”, and “I agree/disagree because . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

Professional basketball players tend to be taller than professional baseball players.

Here are two histograms that show height distributions of 50 male professional baseball players and 50 male professional basketball players.

2. Write 2–3 sentences that describe the distribution of the heights of the basketball players. Comment on the center and spread of the data.
3. Write 2–3 sentences that describe the distribution of the heights of the baseball players. Comment on the center and spread of the data.

### Activity Synthesis

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.

Representing, Writing: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their written responses to the questions about describing the distribution of heights of basketball and baseball players. Ask students to meet with 2–3 partners, to share and get feedback on their responses. Provide listeners with prompts for feedback that will clarify the descriptions about the distributions shown in the histograms (e.g., “What details about the histograms are important are important to describe?”). Finally, give students time to revise their initial written responses to reflect input they received. This will help students write descriptions of the center and spread of distributions displayed in histograms using precise language.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we learn about a different way to represent the distribution of numerical data—using a histogram. Ask students,

Once we have a histogram drawn, we can use it to answer some questions about a data set.

• “How would you describe a typical weight for this group of dogs?” (Dogs in this group typically weigh around 20 to 25 kilograms)
• “What can we say about the spread of the dog weights based on this histogram?” (There is a fairly wide range of weights represented from around 10 kilograms up to a little over 30 kilograms.)
• “In general, what information does a histogram allow us to see? How is it different from a dot plot?” (A bigger picture of the distribution is shown in the histogram, but some of the detail is lost when compared to a dot plot. For example, this histogram does not show the weight of any individual dogs.)

## Student Lesson Summary

### Student Facing

In addition to using dot plots, we can also represent distributions of numerical data using histograms.

Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.