# Lesson 3

A Gallery of Data

## 3.1: Notice and Wonder: Dot Plots (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that distributions can be discussed in terms of shape, which will be useful when students describe data displays in a later activity. While students may notice and wonder many things about these images, shape and the values on the horizontal axis are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that data sets with different values can have distributions with the same shape if all of the values in the data set are increased or decreased by the same value.

### Launch

Display the dot plots for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

The dot plots represent the distribution of the amount of tips, in dollars, left at 2 different restaurants on the same night.

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the shape of the distribution and the values on the horizontal axis of each dot plot do not come up during the conversation, ask students to discuss these ideas using these questions.

- “What do you notice about the shape of each distribution?” (The data is distributed in exactly the same way in each dot plot.)
- “What is the most frequent value in each dot plot?” ($4 and $11)
- “What is the value of the highest tip in each dot plot?” ($10 and $17)
- “What is the value of the lowest tip in each dot plot?” ($1 and $8)
- “What happens if $7 is added to each of the tips in the first dot plot?” (You get the data distribution in the second dot plot.)

## 3.2: Data Displays (25 minutes)

### Activity

In this lesson, students create and display a dot plot and a box plot using numerical data they collected from a survey question in a previous lesson. The focus of the lesson is on creating graphical displays of data, so appropriate data sets take precedence over the class’s actual data. Here are sample data sets for each of the four questions from the previous lesson.

- On average, how many letters are in the family (last) names for students in this class? {4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10}
- On average, what is the furthest, in miles, that each student in this class has ever been from home? {50, 50, 50, 100, 200, 200, 250, 250, 300, 300, 300, 500, 500, 500, 500, 500, 500, 800, 1000, 2000}
- About how long did it take students in this class to get to school this morning? {5, 5, 5, 5, 10, 10, 12, 15, 15, 15, 15, 25, 25, 25, 25, 30, 35, 40, 45, 55}
- On average, how many movies in the theater did each student in the class watch this summer? {0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 6, 6, 7, 7, 8, 10, 12}
- After groups have had a chance to complete their displays, pause the class to discuss how to do a gallery walk. Provide an example display and ask the class to give answers for the questions in the task about the example to help students understand the types of responses expected.

### Launch

Arrange students in groups of 2–4 and assign each group one of the statistical questions for which students already collected numerical data. Provide each group with tools for creating a graphical display. Ask students to pause after the second question for the gallery walk.

*Action and Expression:*Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “Each student in this class….”

*Supports accessibility for: Language; Organization*

### Student Facing

Your teacher will assign your group a statistical question. As a group:

- Create a dot plot, histogram, and box plot to display the distribution of the data.
- Write 3 comments that interpret the data.

As you visit each display, write a sentence or two summarizing the information in the display.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Choose one of the more interesting questions you or a classmate asked and collect data from a larger group, such as more students from the school. Create a data display and compare results from the data collected in class.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may neglect including titles for axes and may forget the importance of building their plots on a number line with equally spaced intervals. Monitor for groups that do not recall the details of making the different types of displays and suggest they refer back to their work in the previous lesson.

### Activity Synthesis

The goal is to make sure students understand how to represent a distribution of data data using a dot plot, histogram, and a box plot, and to interpret each distribution in the context of the data. The purpose of the discussion is to elicit evidence of student thinking about the distributions. Here are some questions for discussion.

- “What are some ways you summarized the information in the display?” (The data were clustered around one value. The dot plot showed this but the box plot did not. The box plot shows the median as a typical value.)
- “If you collected data from all the students in the school, instead of just your classmates, which would you rather create, a dot plot or a box plot? Why?” (A box plot because all you have to do is find the five number summary. In a dot plot, you would have to plot every point and that might be hard to do with the tools that I have).
- “What is the shape of the distribution in your dot plot?” (The data were very spread apart.)
- “What information is displayed by the dot plot that is not displayed by the box plot?” (The dot plot displays all the values in the data set.)
- “What information is displayed by the box plot that is not displayed by the dot plot?” (The box plot displayed the quartiles and the median.)

*Speaking, Writing: MLR 7 Compare and Connect.*Use this routine when students present their graphical displays summarizing the data. Ask students to consider what is the same and what is different about each display. Draw students' attention to the different ways the data are represented (using box plots, histograms, dot plots, etc.) and the benefit of each display for interpreting data. These exchanges can strengthen students' mathematical language use and reasoning to make sense of graphical representations of data.

*Design Principle(s): Maximize meta-awareness: Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students created two different data displays from information they collected in a previous lesson.

- “When you look at the two data displays you made, what information jumps out at you?” (The dot plot shows the shape of the data and it is easy to see the frequency of each value. The box plot shows the median and gives me an idea of the interval that contains the middle fifty percent of the data.)
- “What are some contexts that you have seen dot plots, box plots, or histograms outside of this class?” (I have used dot plots in science class when we collected data from an experiment. I have used histograms when we were reading technical writing in English class.)
- “What do you understand about data displays?” (I can use data displays to show the distribution of data. Different displays allow me to notice information about the distribution of the data in different ways.)

## 3.3: Cool-down - Why Graphical Representations? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

We can represent a distribution of data in several different forms, including lists, dot plots, histograms, and box plots. A list displays all of the values in a data set and can be organized in different ways. This list shows the pH for 30 different water samples.

- 5.9
- 7.6
- 7.5
- 8.2
- 7.6
- 8.6
- 8.1
- 7.9
- 6.1
- 6.3
- 6.9
- 7.1
- 8.4
- 6.5
- 7.2
- 6.8
- 7.3
- 8.1
- 5.8
- 7.5
- 7.1
- 8.4
- 8.0
- 7.2
- 7.4
- 6.5
- 6.8
- 7.0
- 7.4
- 7.6

Here is the same list organized in order from least to greatest.

- 5.8
- 5.9
- 6.1
- 6.3
- 6.5
- 6.5
- 6.8
- 6.8
- 6.9
- 7.0
- 7.1
- 7.1
- 7.2
- 7.2
- 7.3
- 7.4
- 7.4
- 7.5
- 7.5
- 7.6
- 7.6
- 7.6
- 7.9
- 8.0
- 8.1
- 8.1
- 8.2
- 8.4
- 8.4
- 8.6

With the list organized, you can more easily:

- interpret the data
- calculate the values of the five-number summary
- estimate or calculate the mean
- create a dot plot, box plot, or histogram

Here is a dot plot and histogram representing the distribution of the data in the list.

A dot plot is created by putting a dot for each value above the position on a number line. For the pH dot plot, there are 2 water samples with a pH of 6.5 and 1 water sample with a pH of 7. A histogram is made by counting the number of values from the data set in a certain interval and drawing a bar over that interval at a height that matches the count. In the pH histogram, there are 5 water samples that have a pH between 6.5 and 7 (including 6.5, but not 7). Here is a box plot representing the distribution of the same data as the dot plot and histogram.

To create a box plot, you need to find the minimum, first quartile, median, third quartile, and maximum values for the data set. These 5 values are sometimes called the *five-number summary*. Drawing a vertical mark and then connecting the pieces as in the example creates the box plot. For the pH box plot, we can see that the minimum is about 5.8, the median is about 7.4, and the third quartile is around 7.9.