This problem solving activity has a number and algebra (equations and expressions) focus.
Momma’s Pizza Shed has the toppings ham, cheese, salami, chicken, mushrooms, tomato, bacon and pineapple.
Jessie wants 2 toppings on her pizza.
How many choices does she have?
At Pizza Place there are 100 toppings available.
How many choices does Jessie have here?
- Generate linear and quadratic patterns.
- Make predictions using a rule.
- Use a systematic method to solve a problem.
This problem challenges students to find the pattern that exists in the number of combinations of 2 toppings that can be made from a choice of 8 pizza toppings, 100 toppings and t toppings.
Some students may begin by first finding all possible ways of obtaining 2 toppings in the simpler case. However, they should be encouraged to find a more sophisticated way to solve the problem. The approach of counting by listing all possibilities, then counting more systematically is a theme that occurs across mathematics.
See also: Penny’s Pizza, Statistics, Level 4; and Can Stack, Algebra, Level 5.
- Pizza "menus" to stimulate discussion
- Copymaster of the problem (English)
- Copymaster of the problem (Māori)
The Problem
Momma’s Pizza Shed has the toppings ham, cheese, salami, chicken, mushrooms, tomato, bacon and pineapple. Jessie wants 2 toppings on her pizza. How many choices does she have?
At Pizza Place there are 100 toppings available. How many choices does Jessie have here?
Teaching Sequence
- Discuss the students' favourite pizza toppings.
- Talk about how you might choose various toppings if you are only allowed a restricted number.
How many ways are there of choosing 2 toppings from 3 available? - Pose the first part of the problem, and ask students to suggest possible approaches.
- Have the students work in groups on the problem, moving on to the 100 toppings’ problem as appropriate.
- Make the Extension case of t toppings available.
- Have students record solutions as they work.
- As students share their solutions, have them justify the approaches they have taken.
Extension
If a pizza shop has t toppings, in how many ways can Jessie choose 2 toppings?
Solution
Start with the 8 toppings problem.
Ham could be chosen with cheese, salami, chicken, mushrooms, tomato, bacon and pineapple – a total of 7 pairs of toppings.
After having used up ham, cheese could be chosen with salami, chicken, mushrooms, tomato, bacon and pineapple – a total of 6 pairs of toppings.
After having used up ham and cheese, salami could be chosen with chicken, mushrooms, tomato, bacon and pineapple – a total of 5 pairs of toppings.
After having used up ham, cheese and salami- chicken could be chosen with mushrooms, tomato, bacon and pineapple – a total of 4 pairs of toppings.
After having used up ham, cheese, salami and chicken - mushrooms could be chosen with tomato, bacon and pineapple – a total of 3 pairs of toppings.
After having used up ham, cheese, salami, chicken and mushrooms - tomato could be chosen with bacon and pineapple – a total of 2 pairs of toppings.
After having used up ham, cheese, salami, chicken, mushrooms and tomato - bacon could be chosen with pineapple – a total of 1 pair of toppings.
So altogether we have 7 + 6 + 5 + 4 + 3 + 2 + 1 pairs of toppings. This can quickly be added by noting that 7 + 1 = 8, 6 + 2 = 8 and 5 + 3 = 8.
28 is the total.
In choosing 2 toppings from 100, the same strategy can be used. Add:
T = 99 + 98 + 97 + 96 + … + 3 + 2 + 1.
See also:
T = 99 + 98 + 97 + 96 + … + 3 + 2 + 1, and
T = 1 + 2 + 3 + … + 96 + 97 + 98 + 99.
Adding Ts together gives 99 lots of 100 (because 99 + 1 = 100, 98 + 2 = 100, etc.). So
2T = 99 x 100.
Hence T = 99 x 50 = 4950.
Solution to the Extension
2 toppings from t available toppings can be solved:
T = (t – 1) + (t – 2) + … + 2 + 1, or
T = 1 + 2 + … + (t – 2) + (t – 1).
Adding gives 2T = (t – 1) x t.
Hence T = (t – 1)t/2.
Check this out for t = 8 and t = 100.