Lesson 9

The purpose of this warm-up is to prepare students to find the mean of a data set. While the goal of the activity is for students to create an expression with a value close to 4, the discussions should focus on the reasoning and strategies students used in creating their expression. Students should notice that they would get the same result if they divided the value of the entire expression in the numerator by 4 as they would if they divided each number in the numerator by 4 because there are 4 numbers in the numerator. 

During the partner discussions, identify students with different strategies for creating an expression with a value of 4. Ask them to share during the whole-class discussion.

Arrange students in groups of 2. Give students 1 minute of quiet work time, and then 2 minutes to share their response with a partner. Follow with a whole-class discussion. If students do not interpret the directions of “close to 4” to mean that their expression can have an exact value of 4, tell them it can be exactly 4.

Some students may think they need to use all of the digits from 0 to 9. Tell them that only 4 digits need to be used, although they are welcome to try finding a good solution using 2 digit numbers if they want.

Poll the class on whether the value of their expression is exactly 4 or is close to 4. Ask selected students to share their strategy for creating an expression with a value of 4. Record and display their responses for all to see.

As students share their reasoning, consider asking some of the following questions:

• “How did you decide on the value of the numerator?”
• “How did the denominator affect your strategy?”
• “How might your strategy change if the denominator was a different number, say, 6 or 10?”
• “How might your strategy change if the numerator had more numbers or fewer numbers?”

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.  

In this activity, students calculate the mean of a data set and interpret it in the context of the given situation. The first data set students see here has a dozen values, discouraging students from redistributing the values incrementally and encouraging them to use a more efficient method. In the second question, students analyze the values in data sets and use the structure (MP7) to decide whether or not it makes sense that a given mean would match the data set. 

As students work and discuss, notice the reasons they give for why the data sets in the second question could or could not be Tyler's data set. Identify students who recognize that the mean of a data set cannot be expected to be higher or lower than most of the values of the data set, and that a fair-share value would have a value that is roughly in the middle of data values.

Keep students in groups of 2. Give them 6–7 minutes of quiet work time, and then time to discuss their responses with their partner.  

Select a couple of students to share how they found the mean of Mai's travel times. Poll the class briefly to see if others in the class found the mean the same way.

Then, focus the discussion on the second task and on what values could be reasonably expected of a data set with a particular mean. Ask students how they decided to rule out or keep certain sets of data as potentially belonging to Tyler. If not mentioned by students, highlight that the mean of a data set would be a value in the middle of the range of numbers in order for it to be a fair-share value.

Point out that, unlike hours of work or cats in crates, the times of travel here cannot actually be redistributed. The interpretation of mean translates into a thought experiment:

The mean is the travel time each day if all the travel times in the set were the same such that the combined travel time for this data set and the original data set was the same.