Lesson 4

This warm-up encourages students to analyze the structure and value of expressions, and to connect them to the process of calculating a mean.  Each expression has one obvious reason it does not belong, however, there is not one single correct answer.

As students discuss in small groups, listen for ideas related to finding the mean of a data set. Highlight these ideas during whole-class discussion.

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 one expression that does not belong and can explain why. When the minute is up, ask them to share their thinking with their small group, and then, together, find at least one reason each expression doesn't belong.

Ask each group to share one reason why a particular expression does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question, attend to students’ explanations and ensure the reasons given are reasonable. If students give unsubstantiated claims, ask them to substantiate them.

At the end of the discussion, ask students which expression or expressions most represent how they would find the mean of a data set. There is a reason each expression (other than C) could represent how they would find the mean of a data set, however, highlight reasoning about the number of terms in the numerator being the same as the value of the denominator (e.g., there are 5 terms in the numerator and the denominator is 5).

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.

The first set of problems (about cats in crates) can be made even more concrete by providing students with blocks or snap cubes that they can physically distribute into piles or containers.  Students using the digital activities will engage with an applet that allows students to sort cats. For the second set of problems (about hours of work), students are prompted to draw two representations of the number of hours of work before and after they are redistributed, creating a visual representation of fair shares or quantities being leveled out.

As students work, identify those with very different ways of arranging cats into crates 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. 

In the first room, to get each crate 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.

Invite several students with different arrangements of cats in the second room with 6 crates 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 crates and no two crates have the same number of cats. Show that the correct arrangements could redistribute the 18 cats such that there are 3 cats per crate.

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 dividing 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.  

This activity serves two key purposes: to reinforce the idea of the mean as a balance point and a measure of center of a distribution, and to introduce 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. Students also practice calculating mean of a distribution and interpreting it in context. 

Unlike in previous activities, students are given less scaffolding for finding both the mean and the sums of distances from the mean. 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.  

The activity Travel Times (Part 1) is not included in IM 6–8 Math Accelerated.

During the launch, 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.

Keep students in groups of 2. Give students 5 minutes of quiet work time to complete the first set of questions and then 2–3 minutes to discuss their responses with their partner before working on the second set of questions together. 

The term “variation” is used in student text for the first time. If needed, explain that it has a similar meaning as “variability” and refers to how different or alike the data values are.

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 how Lin and Andre’s travel times compare. Display the dot plots of their travel times for all to see.

Discuss:

• “How do the data points in Lin’s dot plot compare to those in Andre’s?”
• “How do their means compare? How do their sums of distances from the mean compare?”
• “What do the sums of distances tell us about the travel times?”
• “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?”

Students should see that larger distances from the mean suggest greater variability in the travel times. Even though both students have the same average travel time (both 14 minutes), Lin’s travel times are much more varied than Andre’s. A couple of Lin’s travel times are a lot longer or shorter than 14 minutes. Overall, her data points are within 6–8 minutes of the mean. For Andre, all of his data points are within 3 minutes of the mean.

The last discussion question prepares students to think about a different way to measure the center of a distribution in upcoming lessons.