In this activity, students are building skills that will help them in mathematical modeling (MP4). They formulate a model of a pizza slice as a sector of a circle. They compute unit costs per square inch of pizza to compare the value of several different vendors’ pizza deals. During the Activity Synthesis, students report their conclusions and the reasoning behind them, and they have an opportunity to consider how to quantify variables beyond price and area.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5). Monitor for groups who choose their pizza based strictly on cost, and those who also consider other variables, such as number of toppings and crust thickness. Here are some approaches students may take, from more common to less common:

• Choose vendor A because it's least expensive per square inch.
• Choose vendor C because it is the largest.
• Choose vendor B because it has the best toppings.
• Choose vendor D because it has thin-crust pizza.

Students may accidentally use the diameter instead of the radius when calculating circle areas. Ask them what expression they are using to calculate circle areas, and to check if they’ve used correct measurements.

Invite previously selected students to share their approaches for choosing a vendor. Sequence the discussion of the approach in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “The photos didn’t show the actual sizes of the pizza slices—the images were scaled-down versions of the real thing. Why could we use them to gather information about the areas of the slices?” (Scaling preserves angle measures, so we could use the photos to find the measures of the sectors’ central angles.)
• “Are there any other variables that affect which vendor you would choose? How could you quantify these variables?” (Students could calculate the volume of the pizza if crust thickness is important to them. They could try to figure out a calorie count if they want the most energy for their money. They could create an expression that assigns relative importance to toppings, size, and crust type.)

The purpose of this discussion is to collect approaches to dividing the pizza and to consider how a pizza with different measurements changes the problem. Ask students to share their strategies for deciding where to divide the pizza slice. Ask students, “What would change if the radius and central angle were different? What would stay the same?” (The process would be the same. But when the angle increases past a certain point, the isosceles triangle, whose congruent sides are radii, starts to have a smaller area than the rest of the sector, making the problem impossible.)

If students aren’t sure how to begin, remind them of the name of the unit: “Circles.” Are there any circles that relate to triangles? Are there any other theorems about circles that might help?

The purpose is to discuss how this problem relates to inscribed angles and other concepts students have recently worked with.

Here are some questions for discussion:

• “What are some vocabulary terms from this unit that are relevant to this problem?” (The sides of the beam of light are represented by chords. We needed to find the circumcenter of the triangle to construct the circumscribed circle. The angle made by the sides of the light beam is an inscribed angle.)
• “How many possible positions are there for the light?” (Technically, there are an infinite number of positions. Any point on the circumscribed circle that is in front of the backdrop will work.)
• “What are some other situations where inscribed angles might be relevant?” (A spotlight shining on a stage is similar. Sightlines and viewing angles in auditoriums might involve inscribed angles. There could be applications in design: creating the spoke pattern of a car wheel or designing a logo for a company.)

Here are some additional questions to ask as students make sense of the large Nazca spiral:

• "What is the radius of the circle?" (about 41 meters)
• "What is the circumference of the circle?" (about 258 meters)
• "What is the area of the circle?" (about 5307 square meters)