## Monday, 5 October 2015

### Expanding three binomials - developing conceptual understanding

I have been doing some work with my NQTs recently on developing their conceptual understanding of maths and looking at how we can ensure that our approaches in one area allow proper generalisation and links to other areas. One of the things we were discussing was the idea of brackets expansion, partly motivated by the need to expand products of 3 brackets in the new GCSE. We started looking at why brackets expand the way they do, using images like this:

and drawing the links between the expansion and area. Logically of course this leads to the idea of volume for 3 brackets expansion, but this led me to a problem (one I was discussing a little on Twitter recently), how does this generalise the approach? Allow me to elaborate with an example

This cuboid could be broken down and so the concept of multiplying all of the parts together can be demonstrated, but as a method for completing the calculation this is inefficient to say the least. So how to generalise a method and retain the conceptual understanding? I was puzzling over this when it hit me - treat the cuboid as a prism and apply the prism volume formula volume = area of cross section x depth.

This may seem like a small thing, but what it allows is a generalisation of methods for multiplying two brackets. Whilst I don't think this could be applied to the FOIL method, certainly applying to the grid method would work:

Now this might seem to be an obvious approach to most people, but the real power of it to me was that I could link it back to the volume, with the first calculation giving the area of the side face, and then this area being multiplied by the third length. My NQTs and I can also apply a direct expansion method; i.e.:

x(x - 2) + 3(x - 2) = x2 + x - 6       then

2x(x2 + x - 6) + 5(x2 + x - 6) = 2x3 + 7x2 - 7x - 30

So why am I blogging this? Well I think it is important that pupils do gain that conceptual understanding that links two factor expansion to area, and then 3 factor expansion to volume, and I think the methods applied to expand products of 2 or 3 brackets should reflect this conceptual understanding. This is the approach I will be using with my Year 10 set 5 of 6 in the coming week, and I think it has a great chance of success.