In this activity, students calculate the remaining measurements for the circles in the previous activity. Different students use different approximations of : 3.14, and 3.1415927. The last approximation for is the level of precision that many calculators show.

The different approximations for lead to different estimates for the missing measurements. During the whole-class discussion, students share their different answers and discuss which approximation is the most useful in these cases (MP6).

The goal of this discussion is for students to consider the appropriate level of precision for reporting measurements of these objects.

First, invite one student who used each approximation of pi to share their answer for the circumference of the wagon wheel. (9.42 ft, 9.4247781 ft, or 9.428571 ft)

Ask questions like:

• “Why are these answers different?” (They used different values for .)
• “Which answer is correct?” (They are all approximations. None of them is exact.)
• “How different are these answers?” (They are very close, within 0.01 of each other. They agree in the first three digits.)
• “Does it make sense to report the circumference of a wagon wheel to 7 decimal places?” (Not really, because the given measurement, 3 ft, didn’t have any decimal places.)
• “Could the wagon wheel actually be measured that precisely?” (No, at least not with normal measurement tools.)

If time permits, repeat some of these questions for the airplane propeller and sliced orange.

The key takeaways are:

• People use different approximations for pi depending on the situation and the precision of the measurement.
• For situations like these where the measurements themselves do not have too much accuracy, either 3.14 or is probably the most appropriate value of to use.
• Using a more precise value for is always acceptable, but the final answer should not be reported with more precision than the measurements had.

Finally, ask students to share their reasoning for the problem about the bug on the propeller blade. Understanding that the bug travels 5 times the circumference of the circle will help prepare students to solve upcoming problems that involve multiple revolutions around a circle.

In this activity, students calculate the circumference of a circle given its radius, and vice versa. It is not necessary for students to memorize the formula . However, it is helpful to recognize that the circumference of a circle is proportional to its radius as well as to its diameter.

The questions in this activity do not use the word “circumference.” Students use quantitative and abstract reasoning as they connect the length of the willow frame to the circumference of the basket in order to solve problems (MP2).

Monitor for students who use these different strategies:

• Use the radius to find the diameter, and then use the diameter to calculate the circumference
• Multiply the radius by to find the circumference directly

The goal of this discussion is to highlight the relationship between radius and circumference. 

Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “Why do the different approaches lead to the same outcome?”
• “How does pi show up in each method?”
• “Are there any benefits or drawbacks to one method compared to another?”

As students share their thinking, highlight these connections:

• Because multiplication is associative and commutative, changing the order in which you multiply several factors doesn’t change the product.
• To find the circumference of a circle given its radius, you need to multiply by . You can do this in two separate steps or in one combined step.

If time permits, consider displaying the equations and for all to see. Ask students to describe what each equation means. Then, demonstrate substituting in place of to get the equation . Ask students to describe what this equation represents. If desired, explain that the standard way to write expressions like this is to put the number first. Rewrite the third equation as .

In this activity, students compute the length of a figure that is composed of two semicircles and two line segments. Students need to determine how to decompose the figure to calculate its length. For the second question, students have to compose several line segments to determine the diameter of the semicircle.

When students decide how to decompose the figure to find its perimeter, they are making sense of problems and persevering in solving them (MP1).

If students have trouble seeing the circle and rectangle that compose the figure, suggest that they draw additional lines to decompose the figure.

Students might apply the formula for the circumference of a circle to find the circumference of the oval-shaped field and track. If this happens, remind students that the track is not in the shape of a circle. Ask if they see a way to form a circle and a rectangle within the space.

Students who are familiar with the fact that this size track is referred to as a 400-meter track may be confused when their answer does not equal 400 meters. Explain that a runner does not run right on the edge of the track and possibly direct the student to look at the extension.

Most of the discussion will occur in small groups. However, the whole class can debrief on the following questions:

• “In what ways did you decompose the figure into different shapes?”
• “Why did you choose a particular approximation for , and what was the resulting answer?”
• “How are different students’ answers related, and are they all reasonable lengths for this situation?”

Students who use more digits for their approximation of may come up with a slightly different answer, such as 398.1. In general, when making calculations, if only an estimate is desired, then using 3.14 for is usually good enough. In a situation like this, where the given measurements are quite precise, it can be worth trying more digits in the expansion of , but it turns out not to make much difference in this case.