This problem solving activity has an algebra focus.
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 the bottom 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.
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?
If the pirates wanted to put another layer of cannon balls on their pile, they would need to lift up the pyramid and put another layer on the bottom. How many cannon balls would there be in this 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.
There would need to be 5 cannon balls in the new layer, if the layer was placed underneath the present bottom layer. The cannonball pyramid would then have 1 + 2 + 3 + 4 + 5 = 15 cannon balls.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/cannon-balls at 8:56pm on the 26th February 2024