The Sum of the First n Square Numbers

 One of the things I like to do as a teacher is provide visual proofs of why certain formulas work. A nice one that I use a lot in high school shows how to find the sum of the first n natural numbers:

I saw a method recently that shows a lovely visual approach to finding the sum of the first n square numbers and decided to share this with the students who come to our after school Math Contest Club. I use linking cubes to help me illustrate this.

First, I show them the sum of the first n square numbers and ask them to imagine that the last term is a general n by n square (and not a 3 by 3 square).

Then I make six copes of this sum:

Then I group them together: 

Then I get ready to put them together in two symmetric groups:

Then I join two of the sums together as shown to make an incomplete cuboid with a base of n by (n + 1):

Then I join the third sum to each of the blocks to create another incomplete cuboid with a height of (n+1):

Then I join these two blocks together to create a cuboid with dimensions n, (n+1) and (2n+1):

A simple division now allows me to get the sum of the first n square numbers:

This is a formula that I learnt to prove by induction but it is only when I have modelled it concretely that I fully understand why it works (especially the '2n+1' part and the '÷6' part).