## Saturday, 20 April 2013

### Factorising: a Divisive Topic

See what I did there?

We had a faculty meeting last week, which can only mean one thing: a raging argument about the best way to teach something. We're an opinionated bunch.

The topic in question was quadratic equations. More specifically, factorising, although we touched on solving them in general.

What method would you use to factorise this?
6x2 + 26x + 20 = 0

The mathematician answer, of course, is I wouldn't, I would just use the quadratic formula.
The more observant amongst you will have divided by two first and then found the answer quite easily.
For the purpose of this exercise, dividing by two is not allowed, and neither is taking two out as a factor in the beginning.

This is the method that was put up on the board for us to discuss:

6x2 + 26x + 20 = 0
20 * 6 = 120
Two factors of 120 which add to make 26 are 20 and 6.
6x2 + 20x + 6x + 20 = 0
2x (3x + 10) + 2(3x + 10) = 0
(2x + 2)(3x + 10) = 0

I would estimate that about a third of our faculty looked at this and immediately said, yes, that's how I teach it. Another third said, I know that method but I don't really teach it, and the last third said, I've never understood that method, I think it's stupid.

OK, that last group was probably smaller than the other two, but it felt bigger because I was in it.

I hate this method! It took me ages to get my head round it. I finally got round to proving it to myself so I feel happier, but there is no way I would teach this to a class, because first I'd have to prove it to them to show them where it comes from. I don't think they'd be able to use this method without understanding how it works.

But apparently, I am wrong. There are a lot of students who are successful using this method, and many teachers swear by it. The alternative, trial and error kind of method that I use is obviously quite annoying (hence why in those cases where the coefficient of the x squared term is not 1 or a prime, I would never bother factorising) and it probably puts some students off.

So here's the big question: should we teach students a method they do not understand if it makes getting the answer faster?

Primary schools have moved away from teaching un-understandable methods like column subtraction, bus-stop division, long multiplication in columns, etc towards using methods that students can understand how they work, like chunking, partitioning, open numberlines, the grid method, etc. Surely we should be doing the same higher up the key stages?

I would love to know your opinions on this. If you're a maths teacher, which method do you teach? If you're not, what method were you taught when you were in school?

Emma x x x

Appendix: Explanation of above method

ax2 + bx + c
= (kx + l)(mx + n)
=kmx2 + knx + lmx + ln
=kmx2 + (kn + lm)x + ln

So we are looking for k, l, m and n such that:
km = a
kn + lm = b
ln = c

Aim to split b up into kn + lm
ac = kmln = kn * lm
So we can factorise ac such that the two factors sum to make b.
Working forwards:
ax2 + bx + c
= kmx2 + knx + lmx + ln
=kx(mx + n) + l(mx + n)
= (kx + l)(mx + n)

1. I had never even heard of that method until yesterday when I asked one of the teachers in my department because I knew I have to teach it next week. He showed me about 3 or 4 different ways of doing it, but I still just think the trial and error way of doing it is the most intuitive. I like how you've explained it in the appendix - if I did end up teaching it that way now I think I would be much happier, but for the moment I'm going to focus on coefficients of 1 in front of the x squared and worry about ax^2 later on.

1. Wow, that was oddly well-timed then!
I'm sticking with trial and error all the way. This method just feels like a hack.

2. I'm in the group that knows about it, but I wasn't as comfortable teaching it as the others.

Although it *looks* a bit hacky, the way that it works is pretty explanatory IMO- because students are already familiar with single bracket factorisation, it reduces a quadratic factorisation to a couple of single bracket factorisations, and with minimal effort.

In contrast, the trial and error method that tend to is probably better received by pupils in upper sets because it helps to broaden their minds on how to do quadratics in different ways rather than one that builds more rigidly on the things that have come before.

3. What about systematic trial and error:

6x^2 + 26x + 20 = 0

Factorise 6x^2 and 20 and list them:

6x 3x . . . . . 20 1 2 10 4 5

x 2x . . . . . 1 20 10 2 5 4

Cross multiply and add between the two separate columns to get 26x

and you will find 2 possible answers

(3x + 10)(2x + 2) or (6x + 20)(x + 1)

1. Yeah, that's exactly how I was taught and that's what I still do, although I tend not to make a little table any more.