## Pizza-Topping Combinations

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?

Let students discuss possible answers.

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 poster board 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.

### Reference

Sharon L. Young, The Arithmetic Teacher, April, 1991, vol. 38, no.8. Pp. 28-33.

- Scissors
- Paste
- Round pieces of red construction paper or cardstock (to represent the pizza)
- Pizza-Topping Combinations Activity Sheet

**Assessment Option**

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**

- Obtain pizzeria menus and have students compute prices for pizzas with various combinations of toppings.
- 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

### Favorite Pizza Toppings

### Learning Objectives

Students will:

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

### NCTM Standards and Expectations

- Predict the probability of outcomes of simple experiments and test the predictions.