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.
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?
Sometimes fruit is piled this way at supermarkets.
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?
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.
If a layer of cannon balls is put underneath the present bottom one it would need 1 + 2 + 3 + 4 + 5 = 15 cannon balls.