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.
In this course, students have not yet been introduced to the term mean. Do not introduce the term at this time. Students could say that expressions A, C, and D go together because the denominator matches the number of terms in the numerator.
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.
Activity
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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.
4.2
Activity
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.
This activity introduces students to the concept of mean or average in terms of equal distribution or fair share. The two contexts chosen are simple and accessible, and include both discrete and continuous values. Diagrams are used to help students visualize the distribution of values into equal amounts.
In the digital version of the activity, students use an applet to rearrange the cats in boxes. The applet allows students to drag images of cats into different boxes to evenly distribute them. Use the digital version if there isn’t enough available physical equipment for students to experiment in small groups, or if time is limited and students can access the applet readily on devices.
As students work, identify those with very different ways of arranging cats into boxes to obtain a mean of 6 cats. Also look for students who determine the redistributed work hours differently. For example, some students may do so by moving the number of hours bit by bit, from a server with the most hours to the one with the fewest hours, and continue to adjust until all servers have the same number. Others may add all the hours and divide the sum by the number of servers.
Activity Synthesis
Invite several students with different arrangements of cats in the second room with 6 boxes to share their solutions and how they know the mean number of cats for their solutions is 3. Make sure everyone understands that their arrangement is correct as long as it had a total of 18 kittens and 6 boxes and no 2 boxes have the same number of cats. Show that the correct arrangements could redistribute the 18 cats such that there are 3 cats per box.
Then, select previously identified students to share how they found the redistributed work hours if the workers were to spread the workload equally. Start with students who reallocated the hours incrementally (from one server to another server) until the hours level out, and then those who added the work hours and divided the sum by 5.
Students should see that the mean can be interpreted as what each member of the group would get if everything is distributed equally, without changing the sum of values.
4.3
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.
Students may need some extra orientation to this activity because “Travel Times (Part 1)” is not included in this course.
During the Launch, explain what is meant by the “sums of distances to the left and right of the mean.” Consider demonstrating how to find these sums for Diego’s travel times. ( and )
During the Activity Synthesis, tell students that a measure of center for a data distribution is a number that can be thought of as the middle or typical value of the distribution. 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.
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?
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
The purpose of this discussion is to talk about students’ understanding of mean. Here are some questions to support the discussion:
“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.)
“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.)
“Why might it make sense to think of the mean as a ‘fair share’?” (Often we think of “fair” as meaning each member of the group gets the same amount. The mean is the amount each member gets when the total is redistributed in this way.)
“How do we find the mean of a data set?” (Add all the values together, and then divide by the number of values.)
Student Lesson Summary
Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use for the center to summarize a distribution.
We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.
For example, suppose there are 5 containers, each of which has a different amount of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.
There are 5 identical tape diagrams that are each partitioned into 4 equal parts. The first diagram has 1 part shaded. The second diagram has 4 parts shaded. The third diagram has 2 parts shaded. The fourth diagram has 3 parts shaded. The fifth diagram has no parts shaded.
To find the mean, first we add up all of the values. We can think of this as putting all of the water together: .
To find the “fair share,” we divide the 10 liters equally into the 5 containers: .
In general, to find the mean of a data set with values, we add all of the values and divide the sum by .
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.
A dot plot for "number of stickers". The numbers 18 through 24 are indicated. There is a triangle indicated at 21 stickers. There are lines indicating the distance between 18 and 21, the distance between 19 and 21, the distance between 21 and 22, 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.
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.
Arrange students in groups of 2. Provide access to straightedges.
Prepare a collection of objects such as snap cubes, stickers, or other desirable objects so that each group could have 10. Give a majority of the objects to 1 group, 1 or 2 objects to a few other groups, and leave most groups with none. Ask students, “What do you think about how I’ve given out these things?” Listen for someone to say that it is “not fair.” Then ask, “How could we pass these out so that it is fair?”
Redistribute the objects using a suggested method so that each group has 10 objects to use for the first question.
Give students 3–4 minutes of quiet work time to complete the first set of questions and 1–2 minutes to share their responses with a partner. Since there are many possible correct responses to the question about the boxes in a second room, consider asking students to convince their partner that the distribution that they came up with indeed has an average of 3 kittens per box. Then, give students 4–5 minutes to work together on the second set of questions.
Action and Expression: Develop Expression and Communication. Give students access to snap cubes or blocks. Supports accessibility for: Conceptual Processing, Organization
Activity
None
The stuffed toy kittens in a preschool room are placed in 5 boxes.
The preschool teacher wants the kittens distributed equally among the boxes. How might that be done? How many kittens will end up in each box?
The number of kittens in each box after they are equally distributed is called the mean number of kittens per box, or the average number of kittens per box. Explain how the expression is related to the average.
Another preschool room has 6 boxes. No 2 boxes have the same number of kittens, and there is an average of 3 kittens per box. Draw or describe at least 2 different arrangements of kittens that match this description.
Five servers are scheduled to work the number of hours shown. They decide to share the workload, so each one would work equal hours.
Server A: 3
Server B: 6
Server C: 11
Server D: 7
Server E: 4
On the first grid, draw 5 bars whose heights represent the hours worked by Servers A, B, C, D, and E.
Then, think about how you would rearrange the hours so that each server gets a fair share. On the second grid, draw a new graph to represent the rearranged hours. Be prepared to explain your reasoning.
Based on your second drawing, what is the average, or mean, number of hours that the servers will work?
Explain why we can also find the mean by finding the value of the expression .
Which server will see the biggest change to work hours? Which server will see the least change?
Student Response
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Building on Student Thinking
In the first room, to get each box to have the same number of cats, some students might add new cats, not realizing that to “distribute equally” means to rearrange and reallocate existing quantities, rather than adding new quantities. Clarify the meaning of the phrase for these students.
Some students may not recognize that the hours for the servers could be divided so as to not be whole numbers. For example, some may try to give 4 servers 6 hours and 1 server has 7 hours. In this case, the time spent working is still not really divided equally, so ask the student to think of dividing the hours among the servers more evenly if possible.
