Unit 8, Lesson 10
Finding and Interpreting the Mean as the Balance Point
Grade 6
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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.
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.
Which three go together? Why do they go together?
A
B
C
D
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:
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.
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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.
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.
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.
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?
Let’s investigate whether 10 can produce similar sums as those of 11.
| 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?
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:
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.
MLR5: Co-Craft Questions
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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.
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.
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.
Here is a dot plot showing lengths of Lin’s trips to school.
| time in minutes | distance from the mean | left or right of the mean? |
|---|---|---|
| 22 | ||
| 18 | ||
| 11 | ||
| 8 | ||
| 11 |
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:
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:
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
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.
Help us improve by sharing suggestions or reporting issues.
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.