This Warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.
Student Lesson in Spanish
Activity
None
Which three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology that they use, such as “numerator,” “denominator,” “mean,” or “sum,“ and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
If it is not mentioned, bring up that options A, C, and D can go together because they could be used to calculate a mean.
10.2
Activity
Standards Alignment
Building On
Addressing
6.SP.B.5.c
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 explore the idea of the mean as a measure of center of all the values in the data, using a dot plot to help them visualize this idea. Students determine the distance between each data point and the mean, and notice that the sum of distances to the left is equal to the sum of distances to the right. In this sense, the mean can been seen as “balancing” the sets of points with smaller values than it and those with larger values. They make use of the structure (MP7) to calculate the distance between each data point and another point that is not the mean to see that the sums on the two sides are not equal. The idea of the mean as a measure of center of a distribution is introduced in this context.
As students work and discuss, identify those who could articulate why the mean can be considered a balancing point of a data set.
Launch
Arrange students in groups of 2. Give students 5 minutes to complete the first two questions with a partner, and then 5 minutes of quiet work time to complete the last two questions. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Within the first 3–5 minutes, check in with students to provide feedback and encouragement. Check to see how they calculate the distance between each point and the mean. If necessary, remind students that distances are positive. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Here are data showing how long it takes for Diego to walk to school, in minutes, over 5 days. The mean number of minutes is 11.
12
7
13
9
14
Represent Diego’s data on a dot plot. Mark the location of the mean with a triangle.
The mean can also be seen as a measure of center that balances the points in a data set. If we find the distance between every point and the mean, add the distances on each side of the mean, and compare the two sums, we can see this balancing.
Record the distance between each point and 11 and its location relative to 11.
time in minutes
distance from 11
left of 11 or right of 11?
12
1
right
7
4
left
13
9
14
Sum of distances left of 11:___________ Sum of distances right of 11:___________
What do you notice about the two sums?
Can another point that is notthe mean produce similar sums of distances?
Let’s investigate whether 10 can produce similar sums as those of 11.
Complete the table with the distance of each data point from 10.
time in minutes
distance from 10
left of 10 or right of 10?
12
7
13
9
14
Sum of distances left of 10:___________ Sum of distances right of 10:___________
What do you notice about the two sums?
Based on your work so far, explain why the mean can be considered a balance point for the data set.
Student Response
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Building on Student Thinking
Some students might write negative values for distances between the mean and points to the left of the mean. They might recall looking at distances between 0 and numbers to the left of it in a previous unit and mistakenly think that numbers to the left of the mean would have a negative distance from the mean. Remind students that distances are always positive—the answer to “How far away?” or “How many units away?” cannot be a negative number.
Activity Synthesis
Select a couple of students to share their observations on the distances between Diego's mean travel time and other points. To facilitate discussion, display this dot plot (with the distances labeled) for all to see. Discuss how the sums of distances change when different points are chosen as a reference from which deviations are measured.
Ask:
“What did you notice about the sums of distances to the mean?” (The sums of the distances to the mean are the same for each side.)
“If you choose another point or location on the number line, would it produce equal sums of distances to the left and to the right?” (No, for any other point, one of the sums will be greater.)
Highlight the idea that only the mean could produce an equal sum of distances. Remind students that they have previously described centers of data sets. Explain that the mean is used as a measure of center for a distribution because it balances the values in a data set. Because data points that are greater than the mean balance with those that are less than the mean, the mean is used to describe what is typical for a data set.
10.3
Activity
Standards Alignment
Building On
Addressing
6.SP.B.5.c
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.
This activity reinforces the idea of the mean as a balance point and a measure of center of a distribution. It also introduces the idea that distances of data points from the mean can help us describe variability in data, which prepares students to think about mean absolute deviation in the next lesson. In addition, students practice both calculating the mean of a distribution and interpreting it in context.
As students work, notice those who may need additional prompts to perform these tasks. Also listen for students’ explanations on what a larger mean tells us in this context. Identify those who can clearly distinguish how the mean differs from deviations from the mean.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Remind students of the context of travel time to school. Use Co-Craft Questions to orient students to the context, and elicit possible mathematical questions.
Display only the problem stem and image of Diego’s and Andre’s dot plots, without revealing the 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 “mean,” “distribution,” and “spread.”
Reveal the questions about the mean and sums of distances to the left and right of the mean, and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
“Is there a main mathematical concept that is present in both your questions and those provided? If so, describe it.”
“How do your questions relate to spread?”
Activity
None
Here are dot plots showing how long Diego’s trips to school took in minutes and how long Andre’s trips to school took in minutes. The dot plots include the means for each data set, and those means are marked by triangles.
A dot plot, Diego's travel time in minutes, 6 to 23 by ones. It has one dot each at 7, 9, 12, 13, 14 and a triangle at 11.
A dot plot, Andre's travel time in minutes, 6 to 23 by ones. It has 2 dots at 12, one dot each at 13, 16, 17, and a triangle at 14.
Which of the two data sets has a larger mean? In this context, what does a larger mean tell us?
Which of the two data sets has larger sums of distances to the left and right of the mean? What do these sums tell us about the variability in Diego’s and Andre’s travel times?
Here is a dot plot showing lengths of Lin’s trips to school.
Calculate the mean of Lin’s travel times.
Complete the table with the distance between each point and the mean as well whether the point is to the left or right of the mean.
time in minutes
distance from the mean
left or right of the mean?
22
18
11
8
11
Find the sum of distances to the left of the mean and the sum of distances to the right of the mean.
Use your work to compare Lin’s travel times to Andre’s. What can you say about their average travel times? What about the variability in their travel times?
Student Response
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Building on Student Thinking
Activity Synthesis
The two big ideas to emphasize during discussion are: what the means tell us in this context, and what the sums of distances to either side of each mean tell us about the travel times.
Select a couple of students to share their analyses of Diego and Andre’s travel times. After each student explains, briefly poll the class for agreement or disagreement. If one or more students disagree with an analysis, ask for their reasoning and alternative explanations.
Then, focus the conversation on how Lin and Andre’s travel times compare. Display the dot plots of their travel times for all to see.
A dot plot, Andre's travel time in minutes, 6 to 23 by ones. It has 2 dots at 12, one dot each at 13, 16, 17, and a triangle at 14.
Discuss:
“How do the data points in Lin’s dot plot compare to those in Andre’s?” (Lin’s dot plot shows the dots much more spread out than the dots in Andre’s dot plot.)
“How do their means compare? How do their sums of distances from the mean compare?” (They have the same mean, but the sum on each side for Andre is 5 while the sum on each side for Lin is 12.)
“What do the sums of distances tell us about the travel times?” (This tells us that Andre’s travel times are more consistent. He often travels around 14 minutes. Lin’s travel times are much more variable.)
“If more than half of Lin’s data points are far from the mean of 14 minutes, is the mean still a good description of her typical travel time? Why or why not?” (Probably not. The mean still gives a central value, but because the times are so variable, it does not give a good idea of what her usual travel is like.)
Lesson Synthesis
In this lesson, we learn that the mean can be interpreted as the balance point of a distribution.
“How does the mean balance the distribution of a data set?” (Thinking of each point on the dot plot as having the same weight, the mean would be where the dot plot is balanced so that the weights are evenly distributed on either side of the mean.)
“How can a dot plot help us make sense of this interpretation?” (Because balance and weight are more natural to think about, it can make it easy to estimate where the balance point is.)
“Could another value—besides the mean—balance a data distribution? How can we tell?” (No. If another point is used, the total distances on the left and right will not be equal.)
We also learn that the mean is used as a measure of center of a distribution, or a number that summarizes the center of a distribution.
“Why might it make sense for the mean to be a number that describes the center of a distribution?” (In order to balance, the mean needs to be somewhere near the center of the distribution.)
“In earlier lessons, we had used an estimate of the center of a distribution to describe what is typical or characteristic of a group. Why might it make sense to use the mean to describe a typical feature of a group?” (The mean is a measure of center, so it could be used as a typical value to describe a distribution.)
Student Lesson Summary
The mean is often used as a measure of center of a distribution. One way to see this is that the mean of a distribution can be seen as the “balance point” for the distribution. Why is this a good way to think about the mean? Let’s look at a very simple set of data on the number of stickers that are on 8 pages:
19
20
20
21
21
22
22
23
Here is a dot plot showing the data set.
Dot plot for “number of stickers”. The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. The data are as follows: 18 stickers, 0 dots. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 2 dots. 23 stickers, 1 dot. 24 stickers, 0 dots.
The distribution shown is completely symmetrical. The mean number of stickers is 21, because . If we mark the location of the mean on the dot plot, we can see that the data points could balance at 21.
In this plot, each point on either side of the mean has a mirror image. For example, the two points at 20 and the two at 22 are the same distance from 21, but each pair is located on either side of 21. We can think of them as balancing each other around 21.
A dot plot for "number of stickers". The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. There are arrows pointing to the dots at 20 and 22. There are lines indicating the distance between 20 and 21 and the distance between 21 and 22. The data are as follows: 18 stickers, 0 dots. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 2 dots. 23 stickers, 1 dot. 24 stickers, 0 dots.
Similarly, the points at 19 and 23 are the same distance from 21 but are on either side of it. They, too, can be seen as balancing each other around 21.
A dot plot for "number of stickers". The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. There are arrows pointing to the dots at 19 and 23. There are lines indicating the distance between 19 and 21 and the distance between 21 and 23. The data are as follows: 18 stickers, 0 dots. 19 stickers, 1 dot. 20 stickers, 2 dots. 21 stickers, 2 dots. 22 stickers, 2 dots. 23 stickers, 1 dot. 24 stickers, 0 dots.
We can say that the distribution of the stickers has a center at 21 because that is its balance point, and that the eight pages, on average, have 21 stickers.
Even when a distribution is not completely symmetrical, the distances of values below the mean, on the whole, balance the distances of values above the mean.
Standards Alignment
Building On
Addressing
6.SP.A.3
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.
