## Pizza-Topping Combinations

3-5
1

Using the pizza topping theme from the previous lesson, students explore the mathematical idea of combinations. Students also discuss whether or not order matters.

Tell students that the cafeteria workers plan to offer pizza with four different toppings: Pepperoni, mushrooms, sausage, and green pepper. Their job is "create" each kind of pizza. If they always put two different toppings on each pizza, how many different pizzas can they make? What combinations are possible?

Distribute copies of the Pizza-Topping Combinations activity sheet to each student. Students will be cutting out pizza "toppings" to use in today's lesson.

 Pizza Toppings Combinations Activity Sheet

Discuss the directions with the students.

Students should work in groups of 3, if possible. First, the students in the group should discuss a systematic way of creating all of the combinations of pizza. Students should use cutout pictures of two different toppings and paste them onto one of the empty pizzas. Continue having the students paste other combinations until all of the combinations have been created.

An interesting discussion point should come up. Does order matter when placing the toppings on the pizza? For example, is pepperoni-mushroom the same as mushroom-pepperoni? Allow time for students to discuss this in their groups and come to a consensus. Remind students that they are using different toppings for their pizzas.

For the purposes of probability, order does not matter when placing toppings on a pizza, so these are the following possible combinations:

Pepperoni and Mushroom
Pepperoni and Sausage
Pepperoni and Green Pepper
Mushroom and Sausage
Mushroom and Green Pepper
Sausage and Green Pepper

After the activity is completed, each group should share their results with the entire class. One group's pizzas can be arranged on posterboard to be "given" to the cafeteria workers.

Ask students how the solution would have been different if toppings could be repeated (e.g. "double" toppings)? Allow students time in their groups to discuss and explore the answer. Students may repeat the previous steps and cut out the topping combinations, or they may use another method to solve this new problem.

The following are the possible combinations when toppings can be repeated:

Pepperoni and Pepperoni
Pepperoni and Mushroom
Pepperoni and Sausage
Pepperoni and Green Pepper
Mushroom and Mushroom
Mushroom and Sausage
Mushroom and Green Pepper
Sausage and Sausage
Sausage and Green Pepper
Green Pepper and Green Pepper

Once again, order does not matter.

Assessments

Students should compare their predictions made at the beginning of the lesson to the final solution discovered. Ask them to explain why their initial prediction was correct or incorrect.

Extensions

1. Obtain pizzeria menus and have students compute prices for pizzas with various combinations of toppings.
2. Ask students how many combinations of two-topping pizzas are possible if toppings can be repeated and if, mathematically, ordered did matter. Based on the counting principle, students should identify 16 combinations. [4 toppings for the first choice × 4 toppings for the second choice.]

Teacher Reflection

• Did students appear to understand why order did not matter for the pizza toppings? Did any students "double-count" toppings?
• What other probability applications can I introduce to students to reinforce the concepts learned in today's lesson?

### Pizza at Home

3-5
Students survey members of their community to determine preferences about pizza. Students compare their results with their classmates and create graphs of their data.

### Favorite Pizza Toppings

3-5
Using a survey, students collect data about favorite pizza toppings. They create a double-bar graph and discuss their graphs with other members of the class.

### Learning Objectives

Students will:

• Explore the mathematical idea of combinations of two items
• Share their findings with fellow classmates