Lesson 14

This warm-up invites students to reason about areas of circles and their sectors in the context of cost-efficiency. This context will be explored further in the next activity.

Ask for 1 or 2 brief responses from students before moving on to the next activity. The important point is that for the same price, the large slice gives more pizza, making it a better deal.

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 students who use protractors, and for those who use the string to measure arc length and radius. Identify groups who choose their pizza based strictly on cost, and those who also consider other variables such as number of toppings and crust thickness.

Arrange students in groups of 4. Distribute the following to each group: one set of cards from the blackline master, protractors, string, rulers, and, if possible, digital tools that can run spreadsheet technology. Tell students that they can choose from any of these tools as they work.

Ask students, “When a restaurant advertises a 12-inch pizza, what does that mean?” (The diameter of the pizza measures 12 inches.)

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.

Select previously identified groups to share their choices in this order: First, choose a group who used string to estimate the angle measures in radians. Next, choose a group who used a protractor to estimate the angle measures in degrees. Ask students to compare and contrast the protractor and string strategies. (String is a more common object and so you’re more likely to have it with you. A protractor gives you the angle measure in one step without needing to calculate a ratio. Both give the same answer, just in different units.)

Then, choose a group who chose vendor A because it represents the least expensive unit cost. Then, choose a group who chose a different vendor based on other variables. Ask students how they could quantify some of these other 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.)

Consider asking students, “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.)

In this activity, students have the opportunity to use what they know about trigonometry and circles to decide how to divide a pizza slice between two people equally so that one of them doesn’t have to eat the crust. Students build skills that will help them in mathematical modeling (MP4) as they decide how to represent the situation. Students may define variables, draw diagrams, and use trial and error.

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

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.)

In this activity, students use properties of circumscribed circles and inscribed angles to locate possible positions for a light in a photography session.

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?

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, when creating the spoke pattern of a car wheel or designing a logo for a company.)