Unit 8, Lesson 4
Dot Plots
Grade 6
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This Math Talk focuses on calculating percentages from parts of a whole. It encourages students to think about percentages and to rely on what they know about division to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students describe how common a value is in a data set as a percentage.
To calculate percentages mentally, students need to look for and make use of structure (MP7). In describing their strategies, students need to be precise in their word choice and use of language (MP6).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Find each value as a percentage.
To involve more students in the conversation, consider asking:
When examining displays of data, it can help to consider frequencies as percentages to determine how often certain values, or ranges of values, appear in the data set.
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In this activity, students use a list of data to complete a frequency table and draw a dot plot. The drawing of the dot plot should be fairly straightforward. The emphasis here is on using a graphical representation to study and comment on the data distribution, and to reinforce how it allows us to make observations that are difficult to make by looking at lists and tables.
As students work, identify those who describe distributions not only in terms of individual categories or values (like “four people did not order any upgrades”) but also characterize them in broader terms (like “almost all customers in the data set ordered two or fewer upgrades”). Invite them to share during the discussion following the activity.
Keep 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.
A computer shop offers upgrades to computers such as better graphic cards, additional memory, a larger monitor, or a wireless mouse. Fifteen customers are asked, “How many upgrades did you add to your computer?” Here are their responses:
Complete the table.
| number of upgrades | frequency (number) |
|---|---|
| 0 | 4 |
| 1 | 5 |
| 2 | |
| 3 |
Use the frequency table to make a dot plot. Label your drawing clearly.
Use your dot plot to study the distribution for the number of upgrades. What do you notice about the number of upgrades that this group of customers ordered? Write 2–3 sentences summarizing your observations.
Display a completed dot plot for all to see. Solicit as many observations or comments about the distribution as time permits. The goal of this discussion is to allow students to hear as many ways to describe distributions as possible. Be sure to select previously identified students to make observations that succinctly capture the distributions.
MLR8: Discussion Supports
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In this activity, students begin to think about how to characterize a distribution as a whole, in terms of its center and spread. Students learn that we can give a general description of a distribution by identifying a location that could be the center of the data, and by noting how alike or different the data points are.
Note that at this point center and spread can only be identified and described informally, via visual observation and intuitive reasoning about how data points are distributed. The lack of precise ways to identify center and spread helps to cultivate the need for more formal measures later.
In this activity, students evaluate and critique another's reasoning (MP3).
Keep students in groups of 2. Ask students to read the question in the stem, “How many hours do you generally spend on homework each week?” Then, ask students to explain why this is a statistical question. This is a statistical question because it can be answered by collecting data, and we can expect variability in the data.
Give them 6–7 minutes of quiet time to work on the first three questions, and then 4–5 minutes to share their responses with their partner and to complete the last questions together. Be sure to leave at least 5 minutes for a whole-class discussion.
Twenty-five sixth-grade students answered the question: “How many hours do you generally spend on homework each week?”
This dot plot shows the number of hours per week that these 25 students reported spending on homework.
Use the dot plot to answer the following questions. For each, show or explain your reasoning.
Ask a couple of students to share their thinking on what number of hours would be a good description of homework time for this group of students and why. Then, select a couple of other students to discuss how alike or different the lengths of homework time are.
Explain that sometimes it helps to describe a set of data generally, or to characterize it as a whole. Ask students to share what they found to be the center of the distribution and whether that was a good description of the set as a whole.
Another way to characterize a data set is by describing its spread, or the variability in the data points. The wider the spread (the more dispersed the data points are on a dot plot), the more variable or different they are. The narrower the spread (the more clustered together), the more alike they are. Ask students:
The purpose of the discussion is to solidify student understanding of typical values for a distribution.
Ask students:
We often collect and analyze data because we are interested in learning what is “typical,” or what is common and can be expected in a group.
Sometimes it is easy to tell what a typical member of the group is. For example, we can say that a typical shape in this set is a large circle.
Just looking at the members of a group doesn’t always tell us what is typical, however. For example, if we are interested in the side length typical of squares in this set, it isn’t easy to do so just by studying the set visually.
In a situation like this, it is helpful to gather the side lengths of the squares in the set and look at their distribution, as shown in this dot plot.
A dot plot for "side lengths in centimeters". The numbers 1 through 8 are indicated. The data are as follows: 2 centimeters, 4 dots. 3 centimeters, 5 dots. 4 centimeters, 3 dots. 5 centimeters, 3 dots. 6 centimeters, 2 dots. 7 centimeters, 1 dot.
We can see that squares with 3 centimeter sides are the most common and many others are about the same size. That means we could say that side lengths of about 3 centimeters are typical of squares in this set.
Help us improve by sharing suggestions or reporting issues.
Students may not recall how to find a percentage. Remind them that a percentage can be found if we know the size of a part and that of a whole. If needed, prompt them to determine the size of the entire data set.