In this lesson, students begin informally connecting center and spread to the word “typical.” They consider a value that could be 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). Students also have a chance to critique the arguments of Clare and Tyler (MP3).
Launch
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.
During the partner discussion, identify students who agree with Clare or Tyler to share during the whole-class discussion.
Activity
None
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 says, “I think we can use 3 kilograms to describe a typical backpack weight of the group because it has the greatest frequency.”
Tyler disagrees and says, “I think 3 kilograms is too low to describe a typical weight. There are some values much greater than 3 and that should make the typical value greater.”
Do you agree with either of them? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
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.
2.2
Activity
Standards Alignment
Building On
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
In this activity, students continue to analyze distributions in terms of where most of the data cluster 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 that they use to talk about the spread, using terms such as “wide,” “narrow,” or “something in between.”
This activity is the first time that students see the dot plot about time spent on homework. The earlier activity referenced in the Task Statement is not included in this course.
Launch
Arrange students in groups of 2. Give students 3–4 minutes of quiet work time for the first three questions, and another 4–5 minutes to share their responses and discuss the last question with a partner.
Students are asked to find a percentage. If necessary, briefly review how to find a percentage.
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.
Activity
None
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.
What percentage of the students reported not talking on the phone?
What percentage of the group reported talking on the phone for 3 hours?
How many hours would you say that these 25 students typically spend talking on the phone?
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.
Student Response
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Building on Student Thinking
Activity Synthesis
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 share the values that they find typical for the number of hours spent on the phone. Ask students how their values relate to what they might consider the center of the distribution.
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 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. There are multiple ways to understand the center of a distribution as well as how the spread can be described. Select a few students to describe the distributions of the two data sets shown in this activity.
2.3
Activity
Standards Alignment
Building On
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
In this activity, students continue to practice finding reasonable values for centers of data and describing variability. The focus 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.
Launch
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.
MLR8 Discussion Supports. Display a sentence frame to support understanding of the context: “A person’s typing speed increases when they type (more or fewer) words per minute because....”. Advances: Conversing, Representing
Activity
None
A keyboarding teacher wonders: “Do typing speeds of students improve after taking a keyboarding course?” Explain why her question is a statistical question.
The teacher records the number of words that her students can type per minute at the beginning of a course and again at the end. The two dot plots show the two data sets.
beginning of course
A dot plot, beginning of course, 8 to 36 by 1's. Number of words per minute. Beginning at 8, number of dots above each increment is 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 3, 1, 2, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0.
end of course
A dot plot, end of course, 8 to 36 by 1's. Number of words per minute. Beginning at 8, number of dots above each increment is 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 0, 0.
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 does not improve. They type 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 because it is near the center of the distribution.
20 words per minute is a good description of the centerof the data set at the end of the course.
There is more variability in the typing speeds at the beginning of the course than at the end, so the students’ typing speeds are more alike at the end.
Overall, how fast would you say that the students type after completing the course? What would you consider the center of the end-of-course data?
Student Response
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Building on Student Thinking
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 group of dots in the second dot plot is overall farther to the right on the number line. Ask them if there is a location on each dot plot around which data points seem to be grouped.
Activity Synthesis
The purpose of the discussion is for students to deepen their understanding of distributions and use 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).
The comparison of distributions is useful for overall trends, but does not describe individual patterns well. For example, there could be a student who typed 30 words per minute at the beginning of the course and 20 words per minute at the end. Overall, the students in the class tended to improve because the center went up, but every student doesn’t necessarily follow the trend.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the distinctions between the distributions such as changes in center or spread, and note connections to the context. Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “What are the center and spread of a distribution? What do they tell us about a typical value for the distribution?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say. . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
Display this dot plot and ask how students see the center and spread in the display.
A dot plot for "cat weights in kilograms". The numbers 2 through 12 are indicated. The data are as follows: 3 kilograms, 2 dots. 3.5 kilograms, 3 dots. 4 kilograms, 4 dots. 4.5 kilograms, 5 dots. 5 kilograms, 5 dots. 5.5 kilograms, 4 dots. 6 kilograms, 3 dots. 6.5 kilograms, 3 dots. 7 kilograms, 1 dot.
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.
A dot plot for "cat weights in kilograms". The numbers 2 through 12 are indicated. The data are as follows: 3 kilograms, 2 dots. 3.5 kilograms, 3 dots. 4 kilograms, 4 dots. 4.5 kilograms, 5 dots. 5 kilograms, 5 dots. 5.5 kilograms, 4 dots. 6 kilograms, 3 dots. 6.5 kilograms, 3 dots. 7 kilograms, 1 dot.
A dot plot for "dog weights in kilograms". The numbers 2 through 12 are indicated. The data are as follows: 5 kilograms, 1 dot. 5.5 kilograms, 2 dots. 6 kilograms, 2 dots. 6.5 kilograms, 3 dots. 7 kilograms, 4 dots. 7.5 kilograms, 4 dots. 8 kilograms, 3 dots. 8.5 kilograms, 3 dots. 9 kilograms, 3 dots. 9.5 kilograms, 2 dots. 10 kilograms, 2 dots. 10.5 kilograms, 1 dot. 11 kilograms, 1 dot.
The collection of points for the cat data is further to the left on the number line than the dog data is. 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 a 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 are. 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 are.
Standards Alignment
Building On
Addressing
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
