In this partner activity, students take turns categorizing problems based on whether the question is related to the circumference or the area of a circle. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Next, each group focuses on one of the first five questions. They estimate appropriate measurements for the context and use these measurements to calculate a reasonable answer (MP2).

Much of the discussion about sorting the cards will have happened in small groups. The goal of this whole-class discussion is for students to articulate how they decide whether an answer is reasonable.

First, have the students who worked on the same question compare answers and strategies. Display these questions to guide their group discussions:

• Did you use the same units?
• How did you come up with your estimate for the size of the circle (radius or diameter)?
• Are your estimates very close? Are they reasonable?
• How did you calculate your answer to the question?
• Are your answers very close? Are they reasonable?

Next, invite students who have different answers to the same question to share their reasoning with the class. For each group, ask the rest of the class “Which of these answers do you think are reasonable? Why?” Make sure students understand that since estimates were called for, there is not one exact correct answer for each of these problems.

Some mistakes that could lead to an unreasonable answer include:

• Making too inaccurate of an initial estimate about the size of the circle.
• Using the diameter as if it is the radius, or vice versa.
• Using the wrong formula for circumference or area.
• Forgetting to address an aspect of the question (such as finding the area of the entire pizza, not one slice).
• Labeling the units incorrectly, like using feet for a measure of area instead of square feet.
• Reporting an answer with more decimals places than is reasonable given the level of precision of the initial estimates.

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Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm “.” It really helped me/my group because .”
• “I picked the norm “.” During the activity, I realized the norm “” would be a better focus because .”

In this activity students create a visual display of the circle problem that they solved previously. They can practice explaining their reasoning more clearly on this display than they did in the previous activity. This gives them an opportunity to organize and record their information in a way that can be shared with others who worked on a different problem. The displays can also serve as a record of reasoning about circles, which can be referred back to later in the year.

Arrange for groups that are assigned the same problem to present their visual displays near one another. Give students a few minutes to visit the displays and to see the estimates that others used to answer the question.

Before students begin a gallery walk, ask them to be prepared to share a couple of observations about how their estimates and strategies are the same as or different from others’. After the gallery walk, invite a couple of students to share their observations.

In this activity students look more closely at the last three situations from the card sort activity (Questions 6 through 8). They analyze and critique two claims about each situation, choosing or supplying the best response and explaining why (MP3).

Students must recognize that in the first situation, one of the claims inaccurately estimates the size of the circle. In the second situation, one of the claims calculates the circumference instead of the area. In the third situation, both claims are inaccurate. One of the claims has the right number but uses square units, and the other has the right units but the wrong number.

A note about interpreting the work of others:

It is not possible to know for certain what Clare or Andre were thinking when they made their calculations. For example, it is likely in the second problem that Clare found the circumference of the cookie instead of its area, but it is not possible to know. A wide range of interpretations need to be considered, always keeping an open mind. 

Students might multiply by a decimal approximation, without recognizing that the answers in the claims are all given in terms of .

Students might not realize that there is an error with both of the claims in the third question.

Finally, students might not realize that they are supposed to analyze the reasonableness of the estimates, not just the mathematical correctness of the calculations.

For each situation, invite groups to share which claim is more accurate and why. To involve more students in the conversation, consider asking questions like:

• “Who can restate ’s reasoning in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”

If it does not come out during the discussion, point out that the formula for the area of a circle has a squared term and that the units of the answer are square units. On the other hand, the formula for the circumference does not have a squared term, and the units of the answer are linear units.

This activity gives students an opportunity to determine and request the information needed to solve problems that involve the circumference and area of circles. The problems have similar contexts to Cards 6 through 8 of the Card Sort activity.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information that 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 that they use and to ask increasingly more precise questions until they get the information that they need (MP6).

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:

• “How did you decide whether to calculate the circumference or the area of the circle?”
• “Some measurements on the data card were given in terms of pi. How did this affect your solving process?”

Highlight for students how given only one measurement of a circle (that is, the radius, diameter, circumference, or area), it is possible to calculate all three of the other measurements.