Unit 8, Lesson 3
Representing Data Graphically
Grade 6
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The purpose of this Warm-up is to give students a chance to generate questions about an image. This leads to a discussion about statistical and non-statistical questions. The context here is used again in another activity, so this also gets students familiar with this situation.
Arrange students in groups of 2. Introduce the context by telling students that Clare collects bottle caps and keeps them in containers. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “statistical questions,” “non-statistical questions,” “numerical data,” and “categorical data.”
MLR8: Discussion Supports
In this activity, students are motivated to create a dot plot by identifying a statistical question and collecting data from the class to answer the question. When the data is initially presented, it is messy and difficult to analyze in an unorganized form. Students are then asked to choose an appropriate representation to use for the data (MP5).
If any groups in the Warm-up asked a statistical question that would require counting the items in the container to answer, use that as a way to transition into this activity.
Tell students to keep their materials closed for this discussion. Give each student a sticky note and a dot sticker. Display the image and questions for all to see.
Give students a minute of quiet think time to determine if each question is a statistical question and their reasoning. Invite students to share their responses. When the class agrees that Question 2 is the only statistical question and has a good reason, ask students to write down on a sticky note an estimate for the number of caps in the jar and to remember their answer. Next, ask them to affix their sticky notes on a wall or a board for all to see, and then use the display to answer the statistical question.
Students should recognize that the data cannot be easily interpreted in this format. Discuss other ways in which the estimates can be displayed so that they can be easily seen and understood. If no students suggest placing the sticky notes along a number line like a dot plot, suggest this idea to them.
Next, look for the smallest and largest estimates (either by asking students or skimming through the sticky notes). Draw or display a number line that spans those two numbers, large enough so that students could affix their dot stickers along it. Have each student put their dot sticker on the right place along the number line. If more than one person made the same estimate, the second person should put theirs higher up on the board in the same horizontal position. When the dot plot is complete, tell students to open their materials and answer the questions.
The purpose of this discussion is to see why a dot plot can be useful to visualize and quickly understand a large amount of data.
Ask students what they noticed about the display and for any questions they had about the display. Ask students what advantages this display might have over other ways of arranging the sticky notes.
Tell students that the actual number of bottle caps in the jars is 82. Ask students how this number relates to the dot plot created by the class.
In general, the average from a group of individual guesses can be more accurate than any individual guess. This phenomenon is called “wisdom of the crowd” and is relied on for things such as reviews for products sold online.
To Gather
Straightedges
In this activity, students create a dot plot on their own. The work of drawing dot plots is not new, but students are asked to describe their analysis of the data broadly. They also learn to use the terms frequency and distribution.
Keep students in groups of 3–4. Display the dot plot for all to see.
Ask students how many cats have 1 spot. (4 cats) Explain that the term frequency associated with a particular item represents the number of times an item occurs in the data set. The value 1 in this data set has a frequency of 4. Ask,
Then ask students to describe how the values are arranged in the dot plot. (There are a lot of cats with only a couple of spots, and then 2 with 4 or more spots.) Explain that the distribution refers to a description of how the data are arranged (or distributed) in the dot plot.
Give groups 5–6 minutes to work together to organize the given data and draw their dot plots. Then, give them 2–3 minutes of quiet time to analyze the dot plots and answer the last two questions, and time for a brief small-group discussion about their responses. Reconvene as a whole class afterward.
As students work and discuss, identify those who are able to describe the distribution of the data clearly and succinctly, as well as students who can articulate why a dot plot is an appropriate representation. Ask them to share later.
Priya wants to know if basketball players on 2 teams have had prior experience in international competitions. She gathers data on the number of times the players were on a team before 2016.
Team 1
3
0
0
0
0
1
0
0
0
0
0
0
Team 2
2
3
3
1
0
2
0
1
1
0
3
1
Organize the information on the two basketball teams into these tables.
Team 1
| number of prior competitions |
frequency (number) |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 |
Team 2
| number of prior competitions |
frequency (number) |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 |
Make a dot plot for each table.
Team 1
Team 2
Study your dot plots. What do they tell you about the competition participation of:
Select previously identified students to share their comments on the competition participation of the two teams and why a dot plot is an appropriate representation. Then, invite them to compare the features and merits of the representations that they used in this lesson. Discuss:
Explain to students that a frequency table, bar graph, and dot plot all tell us about the distribution of a data set: each of them lists or shows all the possible values or categories in a data set and how often each one occurs.
Optional
To Gather
Straightedges
This activity is marked optional because it goes beyond the expectations of the standards. The activity helps students think about how bar graphs are useful and how they compare to dot plots. In later lessons, students examine histograms, and the emphasis on categorical and numerical data here can help distinguish histograms from bar graphs.
In this activity, students organize categorical data into frequency tables and represent them as a bar graph. Neither task is new to the grade level, but students approach data analysis with a new awareness of data types, and they use data to answer questions that are more open-ended. Later, they will contrast the representations for categorical data with those for numerical data. Students also learn to use the term "frequency" with categorical data.
As students draw and analyze their bar graphs, listen for the questions they might ask one another about how to draw the bar graph. For example, they might wonder about the order in which the categories are displayed, the width of the bars, or whether certain sports might belong in the same category (for example, swimming and diving are related sports that could be grouped). Also notice the arguments that students make about whether a dot plot would be a suitable representation for the data.
Keep students in groups of 3–4. Provide access to straightedges.
Display this bar graph for all to see. Tell students that this represents the types of shows that Noah watched over the past few months.
Ask students to share what they know or remember about bar graphs. Then, ask:
Give groups 5–6 minutes to work together to organize the given data and draw their bar graph. Then, give them 2–3 minutes of quiet time to analyze the bar graph and answer the last two questions, and time for a brief small-group discussion about their responses. Reconvene as a whole class afterward.
The purpose of the discussion is for students to recognize when it is appropriate to use a bar graph and how bar graphs compare to dot plots.
Focus the whole-class discussion on the decisions that students made when drawing their bar graph and on the last two questions. Some discussion questions:
Select a couple of students to share their observations about favorite summer sports in Kiran's class. Ask the class to see if they agree with those observations. If they don't find those conclusions to be reasonable, ask for their reasoning and alternative conclusions.
Ask students, “How can you decide whether or not to use a dot plot?” (Dot plots should be used to represent only numerical data along a number line and not categorical data.)
When we analyze data, we are often interested in the distribution, which is information that shows all the data values and how often they occur.
In a previous lesson, we saw data about 10 dogs. We can see the distribution of the dog weights in a table such as this one.
| weight in kilograms | frequency |
|---|---|
| 6 | 1 |
| 7 | 3 |
| 10 | 2 |
| 32 | 1 |
| 35 | 2 |
| 36 | 1 |
The term frequency refers to the number of times a data value occurs. In this case, we see that there are 3 dogs that weigh 7 kilograms, so 3 is the frequency for the value “7 kilograms.”
Recall that dot plots are often used to represent numerical data. Like a frequency table, a dot plot also shows the distribution of a data set. This dot plot shows the distribution of dog weights.
A dot plot for "dog weights in kilograms". The numbers 5 through 40, in increments of 5, are indicated. The data are as follows: 6 kilograms, 1 dot. 7 kilograms, 3 dots. 10 kilograms, 2 dots. 32 kilograms, 1 dot. 35 kilograms, 2 dots. 36 kilograms, 1 dot.
A dot plot uses a horizontal number line. We show the frequency of a value by the number of dots drawn above that value. Here, the two dots above the number 35 tell us that there are two dogs that weigh 35 kilograms.
The distribution of categorical data can also be shown in a table. This table shows the distribution of dog breeds.
| breed | frequency |
|---|---|
| pug | 9 |
| beagle | 9 |
| German shepherd | 12 |
We often represent the distribution of categorical data using a bar graph.
A bar graph. The categories “pugs”, “beagles”, and “German shepherds” are labeled on the horizontal axis. The numbers 0 through 12 are indicated on the vertical axis. The data represented by the bars are as follows: pugs, 9. beagles, 9. German shepherds, 12.
A bar graph also uses a horizontal line. Above it we draw a rectangle (or “bar”) to represent each category in the data set. The height of a bar tells us the frequency of the category. There are 12 German shepherds in the data set, so the bar for this category is 12 units tall. Below the line we write the labels for the categories.
In a dot plot, a data value is placed according to its position on the number line. A weight of 10 kilograms must be shown as a dot above 10 on the number line.
In a bar graph, however, the categories can be listed in any order. The bar that shows the frequency of pugs can be rearranged along the horizontal line.
Help us improve by sharing suggestions or reporting issues.
When drawing dot plots, some students might use dots of different sizes or neglect to stack the dots in a straight column. Remind students to use uniform dots and to stack them vertically, using a straightedge as a guide, if needed. For some students, the use of graph paper may be helpful.
Kiran wants to know which three summer sports are most popular in his class. He surveyed his classmates on their favorite summer sport. Here are their responses.
swimming
gymnastics
track and field
volleyball
swimming
swimming
diving
track and field
gymnastics
basketball
basketball
volleyball
track and field
track and field
volleyball
gymnastics
diving
gymnastics
volleyball
rowing
track and field
track and field
soccer
swimming
gymnastics
track and field
swimming
rowing
diving
soccer
Organize the responses in a table to help him find which summer sports are most popular in his class
| sport | frequency |
|---|---|
Represent the information in the table as a bar graph.
When determining the frequencies of different sports, students might lose track of their counting. If this happens, urge students to check off each sport as they account for it and then to double-check their counts afterward.