Lesson 13

The purpose of this Math Talk is to elicit strategies and understandings for computing values from expressions of the form \(a - 1.5\boldcdot b\). These understandings will be useful in a later lesson when students use expressions like \(\text{Q1} - 1.5 \boldcdot \text{IQR}\) to determine if values are outliers. 

In this activity, students have an opportunity to notice and make use of structure (MP7) because they are building up strategies that will enable them to both multiply by 1.5 and subtract that answer from another value.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

This is the first info gap activity in the course. See the launch for extended instructions for facilitating this activity successfully.

This info gap activity gives students an opportunity to determine and request the information needed to investigate and interpret measures of center and variability. 

The information gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

This is the first time in this course that students do the information gap cards instructional routine, so it is important to demonstrate the routine in a whole-class discussion before they do the routine with each other.

Explain the info gap routine: students work with a partner. One partner gets a problem card with a math question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information.. Once the partner with the problem card has enough information to solve the problem, both partners can look at the problem card and solve the problem independently. This graphic illustrates a framework for the routine:

Tell students that first, a demonstration will be conducted with the whole class. As a class, they are playing the role of the person with the problem card while you play the role of the person with the data card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question.

Display this problem for all to see: 

100 captive Asian elephants and 100 wild Asian elephants are weighed. Is there a significant difference between the weights of the two groups of elephants? Explain your reasoning.

Explain that you will be playing the role of the person with the data card. Ask students, “What specific information do you need to find out if there is a significant difference between the weight of captive and wild Asian elephants?” Select students to ask their questions. Respond to each question with, “Why do you need that information?” Here is information from the data card. After this point, only answer the question if these data are relevant:

Data Card

• The distribution of weights for each set of elephants is approximately symmetric.

Captive Asian elephants:

• Mean weight of captive elephants: 3,073 kg
• Standard deviation of captive elephant weights: 282 kg
• Median weight of captive elephants: 3,055 kg
• IQR of captive elephant weights: 399 kg

Wild Asian elephants:

• Mean weight of wild elephants: 2,373 kg
• Standard deviation of wild elephant weights: 121 kg
• Median weight of wild elephants: 2,386 kg
• IQR of wild elephant weights: 163 kg

Explain that if the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, “I don’t have that information.” Ask students to explain to their partner (you) how they used the information to solve the problem. (There is a significant difference in weights between captive and wild Asian elephants. Since the distributions are symmetric, it makes sense to use the mean and standard deviation weights for the two groups. The difference in means of 700 kilograms is very different, even in light of the standard deviations for the groups.)

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What was interesting about Problem Card 1?” (It was interesting to use a real context to think about variability.)
• “Was it more appropriate to use the mean or the median to compare the typical weights for Problem Card 1? Why?” (The mean because of the symmetry of the data.)
• “Which measure of variability did you choose to compare for Problem Card 1? Why?” (The standard deviation, because I used the mean and the data has a symmetric distribution.)
• “What was challenging about Problem Card 2?” (It was difficult to figure out how to make the dot plot without the actual data).
• “How do you know your dot plot is correct?” (There is no way to know. It can only be approximated based on the mean, standard deviation, and the approximate shape.)

Highlight for students how they used the information about the center, shape and variability of the distribution to solve the problems.

This task is provided for optional, additional practice for understanding variability in context.

The mathematical purpose of this activity is to compare data sets, and to interpret the measures of center and measures of variability in the context of the problem.

Keep students in the same groups. Give students 5 minutes to work on the questions and follow with a whole-class discussion.

Identify students who struggle making the connections between the summary statistics and the problem context. Ask them what information the standard deviation conveys. Tell them to focus less on the actual value of the standard deviation and more on which of the two distributions have a greater standard deviation and what that might mean.

The purpose of this discussion is for students to explain the mean and the standard deviation in context.

For each set of graphs, select students read their answers for all three prompts. When necessary, prompt students to revise their language to include the terms shape, measure of center, and variability.