Pizzas often come in three sizes: large, medium and personal. As you change the size of a pizza, how does the amount of cheesy, delicious inside compare to the amount of crust…and is it ever a good idea to buy the smallest size?

In this lesson, students use the formula for the area of a circle to calculate how much of a pizza is actually pizza, and how much is crust. They write linear and quadratic equations for these functions in terms of the radius (or diameter) of the pizza, and also express these areas as percents of the whole pie’s area. Finally, they talk about how they might use their newfound pizza knowledge to make the best decision when they want some pi(e).

Students will

Model pizzas as circles, and calculate total, inside, and crust areas for three different pizza sizes

Use the area of a circle to write quadratic functions for the total and inside areas of a pizza with any radius

Write a linear function to model the crust area of a pizza with any radius and 1-inch crust thickness

Apply the quadratic formula to determine when the crust area of a pizza equals the non-crust area

Use substitution to rewrite area functions in terms of diameter instead of radius

Write and graph rational functions representing the percent of a pizza that’s crust vs. non-crust

Discuss which pizza size to order based on the percent of the pizza that will be crust vs. non-crust

Before you begin

Students should know the formula for the area of a circle (A = πr^{2}) and be able to use the formula to solve problems. They should also have some familiarity with the quadratic formula, and be able to apply it to find the roots of a quadratic equation. The last question of the lesson has students graph two rational functions, but graphing technology can be used if students don’t have much prior experience here.

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