standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Warm-up
This warm-up reinforces students’ understanding about the relationship between the mean absolute deviation (MAD) and the spread of data. In the given scenarios, the number of people attending the two events and their mean age are the same, but the MADs are different. In the first question, students interpret these measures in the context of the situations. In the second, they draw a dot plot that could represent an age distribution with the same mean and yet another MAD.
As students work and discuss, identify several students who drew dot plots that correctly meet the criteria in the second question. Ask students with different dot plots to share during whole-class discussion.
Students may need more time to make sense of how to generate their own dot plot for the second question. If it is not possible to give students additional time, consider presenting the second question at a different time.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time for the first question, and then 2–3 minutes to work on the second question with a partner. Display the following questions for all to see. Ask students to think about and discuss them before drawing their dot plots:
“How many data points should be on the dot plot?”
“How would the mean help us place the data points?”
“How would the MAD help us place the data points?”
“How should our dot plot compare to the dot plots of data sets A and B?”
Student Facing
Here are two dot plots and two stories. Match each story with a dot plot that could represent it. Be prepared to explain your reasoning.
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Twenty people—high school students, teachers, and invited guests—attended a rehearsal for a high school musical. The mean age was 38.5 years and the MAD was 16.5 years.
High school soccer team practice is usually watched by supporters of the players. One evening, twenty people watched the team practice. The mean age was 38.5 years and the MAD was 12.7 years.
Another evening, twenty people watched the soccer team practice. The mean age was similar to that from the first evening, but the MAD was greater (about 20 years).
Make a dot plot that could illustrate the distribution of ages in this story.
[IMG]
Student Response
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Activity Synthesis
Poll students on their response to the first question. Ask a student to explain how they matched one context to its dot plot and another student to explain the second matching context and dot plot. Record and display their responses for all to see. If possible, record their responses directly on the dot plots.
Ask selected students to share their dot plots for the second question and their reasoning. To involve more students in the conversation, consider asking some of the following questions:
“What was the first piece of information you used to draw your dot plot? Why?”
“How did you decide where to place your dots?”
“How is your dot plot the same or different than the first evening of soccer practice?”
“Do you agree or disagree with this representation of the context? Why?”
“Do you have any questions to ask the student who drew the dot plot?”


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