Pizza Toppings 2
Generate linear and quadratic patterns
Make predictions using a rule
Use a systematic method to solve a problem
This problem links up two apparently different themes that occur in Penny’s Pizza, Statistics, Level 4, Can Stack, Algebra, Level 5. It would certainly help the students in your class if they have tackled the first two of these problems or something like them, before trying this problem.
The aim of this problem is to find the pattern that exists in the number of ways of choosing 2 toppings from a given number, t, say. This is completed in the Extension. Consequently, we don’t expect all of the class to be able to obtain the general answer in terms of t. However, we do expect most of the students to be able to see how to obtain the answer in specific cases, even if the cases involve as many as 100 toppings.
We felt that one way that the students at Level 4 could solve Penny’s Pizza, was to write out all possible ways of obtaining 2 toppings. This may still be necessary for some of your students. However, they should be encouraged to find a more sophisticated way to solve the problem. One way this might be done is to follow the sort of reasoning that was used in the Can Stack.
What we see here, then, is the development of a theme leading from counting by listing all possibilities, to counting more systematically. This kind of development is common in mathematics and indeed is the way that progress in this subject is made. Problems may first be solved in very simple ways. Then someone sees how to do them by a more clever approach. Then someone else is able to apply this new method in another setting. After that, someone else takes things even further still. The later developments may look nothing like the starting point from which everything sprang.
Momma’s Pizza Shed has the toppings ham, cheese, salami, chicken, mushrooms, tomato, bacon and pineapple. Jessie decides that she will have 2 toppings on her pizza. How many choices does she have?
At the Pizza Place At The Centre Of The Universe, there are 100 toppings available. How many choices would Jessie have at this Place?
- Discuss what are the students’s favourite take away foods.
Has anyone been past a fast food place recently?
What toppings do you like on your pizzas?
- 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?
- Talk about the first part of the problem. How do you think that you might do it?
- Let the students go off and work in their groups. The faster groups could be asked to try the 100 toppings’ problem. When they have got on top of that they could then try the case of t toppings.
- Get the groups to report back to the whole class.
- Allow the students time to write up their solutions in their books.
Extension to the problem
If there the shop has t toppings, in how many ways can Jessie choose 2 toppings?
We’ll start off with the 8 toppings problem. Now, 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. So we get 28 as the answer.
If we are facing a choice of 2 toppings from 100, we can adopt the same strategy. This time we’ll have to add up
T = 99 + 98 + 97 + 96 + … + 3 + 2 + 1.
The problem Can Stacks shows us exactly how to do this. We show another way here.
T = 99 + 98 + 97 + 96 + … + 3 + 2 + 1, and
T = 1 + 2 + 3 + … + 96 + 97 + 98 + 99.
If we add these Ts together we get 99 lots of 100 (because 99 + 1 = 100, 98 + 2 = 100, etc.). So
2T = 99 x 100.
Hence T = 99 x 50 = 4950. That should keep you happy for a while.
Solution to the extension
The same argument here gives the case for 2 toppings from t available toppings. The number of ways this can be done is
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.
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|Pizza Toppings 2.pdf||47.21 KB|