There is a pyramid of cannon balls on a pirate ship. The first layer looks like this when you look down on it from above.

How many cannon balls are there in this layer (the first layer)?

How many cannon balls will there be in the second layer?

How many cannon balls will there be in the third layer?

How many cannon balls in the top layer?

How many cannon balls do you need to complete the pyramid?

In this problem students further develop 'a feel for' 2- and 3-dimensional objects. The problem introduces and explores triangular numbers, as the cannon balls in each layer of the pyramid form an equilateral triangle.

In order for the cannon balls to sit on top of each other, the students need to see that one ball will comfortably fit on top of three others. This is best modelled this using tennis balls or oranges.

Triangular numbers become of more interest in higher levels when students explore square numbers, pentagonal numbers and so on. The pictures below show why these numbers are named after geometric objects.

The first three triangular numbers The first three square numbers

In the secondary school, triangular numbers are part of the family of Binomial Coefficients. These numbers have a major part to play in counting, and are vital to probability and statistics generally.

Copymaster of the problem (English)

Tennis balls or other suitable balls to build the pyramid. (Dough could also be used for this.)

### Problem

There is a pyramid of cannon balls on a pirate ship. The first layer looks like this when you look down on it from above.

How many cannon balls are there in this layer (the first layer)?

How many cannon balls will there be in the second layer?

How many cannon balls will there be in the third layer?

How many cannon balls in the top layer?

How many cannon balls do you need to complete the pyramid?

### Teaching Sequence

- Introduce the problem using a number of cannon balls (tennis balls or similar). Ask the students to think of ways that they could stack the balls.
- Read problem. Check that the students understand the meaning of the word 'layers' and also know how the pirates piled up their cannon balls.
- Ask the students to guess how many cannon balls they will need. Record the estimates to check against later.
- Brainstorm for ways to solve the problem. (Link these to problems that they have solved before.)
*What strategies could you use?*

What equipment will you need?

How will you record your information?

What do you have to find out? - As the students work ask questions that focus on the patterns they are using to solve the problem.
*What can you tell me about the cannon balls?*

How are you keeping track of the number of cannon balls? - Share solutions

#### Other contexts

Sometimes fruit is piled this way at supermarkets.

#### Extension to the problem

If the pirates wanted to put another layer of cannon balls on their pile they would need to lift it up and put another triangle on the bottom. How many cannon balls would there be in such a layer?

### Solution

In the first layer there are 1 (across the top) + 2 and 3 (across the middle two rows) + 4 (four at the bottom) = 10 cannon balls.

In the second layer there are 1 + 2 + 3 = 6 cannon balls.

In the third layer there are 1 + 2 = 3 cannon balls.

There is only one cannon ball in the top layer.

All together there are 10 + 6 + 3 + 1 = 20 cannon balls.

#### Solution to the extension:

If a layer of cannon balls is put underneath the present bottom one it would need 1 + 2 + 3 + 4 + 5 = 15 cannon balls.