Square NumbersTask 111 ... Years 4  10SummaryLimited numbers of coloured cubes are used to help students discover that each square number is formed from the previous one by adding an odd number of cubes. This relationship allows prediction of the nth square number, which means that the value of any square number can now be found in at least two ways.This cameo has a From The Classroom section which includes an extension for Year 7 and a look at proof by mathematical induction for senior students. Square Numbers also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The extra challenge in this puzzle investigates the difference between two squares. 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
The card provides guidance for finding answers that are recorded on the sheet. Answers can vary. For example, the openness of the question giving the answer as 36 and asking for the corresponding question might yield:
The 20th square number could also be answered in several ways:

Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
With a good collection of wooden blocks, or linking cubes, and students working in pairs, the outline above provides the content for a rich lesson. Teachers with Poly Plug will be easily able to convert the investigation to this material and if the school is a Calculating Changes member, the activity Squares & Square Roots will enrich the lesson further. With either material, you might like to begin the lesson at a floor space using cards about 20cm x 20cm to create the atmosphere of a learning community and to clarify the problem before the concrete materials are used to deepen the investigation. At this stage, Square Numbers does not have a matching lesson on Maths300, however, Lesson 12, Gauss Beats The Teacher, explores several handson ways to sum number series. Visit Square Numbers on Poly Plug & Tasks. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Square Numbers task is an integral part of:

MacKillop College
Damian Howison 
I read your recent cameo for Square Numbers. It came at a good time because one young lady took to the task but we weren't sure at the end of it where to go. Then I read your cameo the very next day and we came up with something nice.
Damian offers this interesting interpretation of the problem here for all to share. 
Doug Williams Consultant 
From a senior school perspective, this problem could be revisited from the point of view of proof. None of the explanations above is a proof. They are demonstrations based on examples, but no matter how many examples may be successful, this is not a proof that all examples will be successful. Nonetheless such demonstrations can point the way to an hypothesis, in this case, that the sum of the first n odd numbers is n^{2}.
So we have to prove that: 1 + 3 + 5 + ... + (2n  1) = n^{2}and Mathematical Induction is one way to go about this: 
First test the hypothesis for n = 1.Think of this procedure as like climbing an infinite ladder. To guarantee reaching the top you must:
L.H.S = 1
R.H.S = 1^{2} = 1 = L.H.SNow assume the hypothesis is true for k, k less than n, and prove it true for (k + 1).
So assume:
1 + 3 + 5 + ... + (2k  1) = k^{2} ... Equation A
is true and, as a consequence, prove that:
1 + 3 + 5 + ... + (2k  1) + (2[k + 1]  1) = (k + 1)^{2}L.H.S
= 1 + 3 + 5 + ... + (2k  1) + (2[k + 1]  1)
= 1 + 3 + 5 + ... + (2k  1) + (2k + 2  1)
= 1 + 3 + 5 + ... + (2k  1) + (2k +1)
= k^{2} + 2k +1 ... by substituting from equation A
= (k + 1) (k + 1)
= (k+ 1)^{2}
= R.H.S as required.