Thinking Outside the 'Box'

I'm a big fan of the array or 'box' method for multiplication (as I blogged earlier here.)  A twitter chat with Britnny Schjolin last week raised this troubling point however:

I know that many of my colleagues are also impressed by it even though they, like me, might not have see it when they were students. I have also worked with colleagues who have openly stated that the 'box' method is not the proper way to show your work or that they don't like it so they won't show this to their students. I'm not sure how widespread such attitudes are but I honestly feel that we are doing our students a huge disservice by not showing them such a powerful representation that allows for so many different connections to be made. If this means that we as teachers should learn something new, then so be it: as educators we must always be prepared to learn new things.

To show how useful I have found this, here is how I recently solved a problem that I came across by using arrays or the 'box' method. 

Prove that the product of four consecutive numbers is always one less than a perfect square.

I started pretty conventionally by trying to generalise the product of four consecutive numbers:

Well, I don't fancy working out that product, but I know if I rewrite the four consecutive numbers like so:

Now I can rearrange to make use of the difference of squares to make something a little more delightful:

 A quick array is drawn to help me work out this product:

Since I have to prove that the product is one less than a perfect square then I need to consider this:

I am good at factoring quadratics by inspection but not so good with quartics! However, I now decide to draw a square array to help me factor by working out the components of each side. The first part solves itself:

To get the 2a³ term, I need to split this symmetrically across the square and think what the next component must be. This makes things very clear:

 This also helps me get the middle product:

 Now to get the -a² term, I need to have a -a² in each of the top right and bottom left cells:

This immediately gives me the last component from which I can write the term as a perfect square:

Thus the product of four consecutive numbers is always one less than a perfect square.

Now, before I learned about the array or 'box' method, I would have chugged through with the algebra and probably would have eventually reached the same conclusion. However, now I can attack and solve such problems in a fraction of the time and with more clarity. This is why we must teach this method:

It is an incredible mathematical tool.

It is not a new idea either. Recently, for fun, I have decided to work my way through Silvanus P. Thompson's classic Calculus Made Easy and I came across this:

Array models were being used back in 1910!