2.1: Packs on Backs (5 minutes)
In a previous lesson, students were exposed to the ideas of center and spread. Here, they begin connecting that idea informally to the word “typical” and a value that could be considered typical or characteristic of a data set by thinking about two good options and reasonings. They continue to interpret a dot plot in the context of a situation (MP2).
During the partner discussion, identify two students—one who agrees with Clare and another who agrees with Tyler—to share during the whole-class discussion (MP3).
Arrange students in groups of 2. Give students 2 minutes of quiet work time and a minute to share their responses with a partner. Follow with a whole-class discussion.
This dot plot shows the weights of backpacks, in kilograms, of 50 sixth-grade students at a school in New Zealand.
- The dot plot shows several dots at 0 kilograms. What could a value of 0 mean in this context?
Clare and Tyler studied the dot plot.
- Clare said, “I think we can use 3 kilograms to describe a typical backpack weight of the group because it represents 20%—or the largest portion—of the data.”
Tyler disagreed and said, “I think 3 kilograms is too low to describe a typical weight. Half of the dots are for backpacks that are heavier than 3 kilograms, so I would use a larger value.”
Do you agree with either of them? Explain your reasoning.
Ask students to share their response to the first question about data points. Record and display their responses for all to see. Ask the selected students—one who agrees with Clare and another who agrees with Tyler—to share their reasoning. Ask if anyone disagrees with both students, and if so, what value they would consider a better description of the center of the data.
Students should have a reasonable explanation for each argument they favor, but it is not necessary to confirm one way or another at this point. Tell students that we will look more closely at different ways to determine a value that is characteristic of a data set in upcoming activities.
2.2: On the Phone (15 minutes)
Earlier, in the backpack example, students saw a distribution described in terms of where data points are clustered on a dot plot and which values have a large number of occurrences. The shape of that distribution was approximately symmetric. In this activity, they continue to analyze distributions in those terms, and try to identify and interpret the center and spread of a distribution that is not symmetric. The two distributions used here allow students to contrast a narrow spread and a wide spread and develop a deeper understanding of variability.
As students work, notice how students identify a general location for the center of a data set and the descriptions they use to talk about the spread (e.g., “wide,” “narrow,” or “something in between”). Identify students who connect the size of a spread to how different or alike the data points are; ask them to share later. Additionally, identify students who measure spread as the range of the entire data as well as those who use the distance to the center.
Arrange students in groups of 2. Give students 5–6 minutes of quiet work time for the first three sets of questions, and another 4–5 minutes to share their responses and then discuss the last two questions with a partner.
Students are asked to find a percentage. If necessary, briefly review how to find a percentage.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Maximize meta-awareness; Support sense-making
Twenty-five sixth-grade students were asked to estimate how many hours a week they spend talking on the phone. This dot plot represents their reported number of hours of phone usage per week.
- How many of the students reported not talking on the phone during the week? Explain how you know.
- What percentage of the students reported not talking on the phone?
- What is the largest number of hours a student spent talking on the phone per week?
- What percentage of the group reported talking on the phone for this amount of time?
- How many hours would you say that these students typically spend talking on the phone?
- How many minutes per day would that be?
- How would you describe the spread of the data? Would you consider these students’ amounts of time on the phone to be alike or different? Explain your reasoning.
- Here is the dot plot from an earlier activity. It shows the number of hours per week the same group of 25 sixth-grade students reported spending on homework.
Overall, are these students more alike in the amount of time they spend talking on the phone or in the amount of time they spend on homework? Explain your reasoning.
- Suppose someone claimed that these sixth-grade students spend too much time on the phone. Do you agree? Use your analysis of the dot plot to support your answer.
Students may neglect to change the rate given (from minutes per week to hours per week, or to minutes per day) and may draw incorrect conclusions as a result. Ask them to think about the unit they are using in their responses.
The purpose of the discussion is to help students find good ways to describe a distribution based on center and spread.
Select a few students to answer the questions about the first dot plot. Tell students that the “typical” value for the data is generally considered the center. Ask, “What would you consider the center for the two dot plots shown in this activity?”
Ask students how they thought about the spread of the data. If possible, select students who thought of spread as the range of the entire data and those who thought of it as an interval around the center. Ask students to share their interpretation of the what the spread means in the context of using the phone. Make sure to include previously identified students who connect spread to how alike or different the data points are.
Tell students that distributions are generally described using the center and spread. Select a few students to describe the distributions of the two data sets shown in this activity.
2.3: Click-Clack (15 minutes)
Previously students analyzed distributions to identify center and spread. In this activity, they continue to practice finding reasonable values for centers of data and describing variability. The focus, however, is on making use of the structure of distributions (MP7) to compare groups in those terms and interpreting their analyses in the context of a situation (MP2).
By comparing distributions, seeing how center and spread for the same population could change, and making sense of what these changes mean, students deepen their understanding of these concepts before learning about more formal measures of center and variability.
Give students a brief overview on keyboarding courses. Explain that these are classes designed to help people improve their typing speed and accuracy, which they may need for their jobs. Typing proficiency is usually measured in terms of number of words typed per minute; the more words typed correctly per minute, the faster or more proficient one's typing is.
Keep students in groups of 2. Give them 5–6 minutes of quiet time to work on the first two questions, and then 2–3 minutes to discuss their responses and complete the last question together.
Design Principle(s): Support sense-making; Cultivate conversation
- A keyboarding teacher wondered: “Do typing speeds of students improve after taking a keyboarding course?” Explain why her question is a statistical question.
- The teacher recorded the number of words that her students could type per minute at the beginning of a course and again at the end. The two dot plots show the two data sets.
Based on the dot plots, do you agree with each of the following statements about this group of students? Be prepared to explain your reasoning.
Overall, the students’ typing speed did not improve. They typed at the same speed at the end of the course as they did at the beginning.
20 words per minute is a good estimate for how fast, in general, the students typed at the beginning of the course.
20 words per minute is a good description of the center of the data set at the end of the course.
There was more variability in the typing speeds at the beginning of the course than at the end, so the students’ typing speeds were more alike at the end.
Overall, how fast would you say that the students typed after completing the course? What would you consider the center of the end-of-course data?
Are you ready for more?
Use one of these suggestions (or make up your own). Research to create a dot plot with at least 10 values. Then, describe the center and spread of the distribution.
- Points scored by your favorite sports team in its last 10 games
- Length of your 10 favorite movies (in minutes)
- Ages of your favorite 10 celebrities
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Some students might find it challenging to tell where the center of a distribution could be just by looking at a single dot plot. The idea of center might be more apparent when presented in comparative terms. For example, ask them to describe in their own words how the distribution of the first dot plot differs from that of the second dot plot. Students are likely able to say that, compared to the first dot plot, the aggregation of dots in the second dot plot is overall higher on the number line. Ask them if there is a location on each dot plot around which data points seem to congregate.
The purpose of the discussion is for students to deepen their understanding of distributions and using the descriptions to compare two groups.
Focus the whole-class discussion on two ideas:
- The distinctions between the two distributions: Students should see that, overall, the cluster of data points have both shifted up toward a greater number of words per minute (moving its center up) and become more compressed in its spread by the end of the course. Because the center moved up in location, the value we use to describe that center would also increase.
- What the changes in the center and spread tell us in this situation: Students should recognize that a higher center means that, overall, the group has improved in their typing speed. They should see that a narrower spread at the end of the course suggests that there's now less variability in the typing speeds of different students (compared to a much larger variability initially).
Supports accessibility for: Visual-spatial processing
In this lesson, we talk about using the center and the spread of a distribution to describe a data set.
- “What do we mean by the ‘center’ of a distribution?”
- “What do we mean by the ‘spread’ of a distribution?”
- “How does the center and spread of a distribution relate to a typical value that could represent the group?”
- “How can we see center and spread in a dot plot like this?”
2.4: Cool-down - Packing Tomatoes (5 minutes)
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Student Lesson Summary
One way to describe what is typical or characteristic for a data set is by looking at the center and spread of its distribution.
Let’s compare the distribution of cat weights and dog weights shown on these dot plots.
The collection of points for the cat data is further to the left on the number line than the dog data. Based on the dot plots, we may describe the center of the distribution for cat weights to be between 4 and 5 kilograms and the center for dog weights to be between 7 and 8 kilograms.
We often say that values at or near the center of a distribution are typical for that group. This means that a weight of 4–5 kilograms is typical for a cat in the data set, and weight of 7–8 kilograms is typical for a dog.
We also see that the dog weights are more spread out than the cat weights. The difference between the heaviest and lightest cats is only 4 kilograms, but the difference between the heaviest and lightest dogs is 6 kilograms.
A distribution with greater spread tells us that the data have greater variability. In this case, we could say that the cats are more similar in their weights than the dogs.
In future lessons, we will discuss how to measure the center and spread of a distribution.