Factorising Trickier Quadratics
Written by Joanne Morgan Wednesday, 12 September 2018
'How to Solve An Adfected Quadratic' is a workshop being run by Joanne Morgan at #MathsConf17.
Factorising a quadratic when the coefficient of x 2 doesn't equal one (a 'non-monic') is apparently one of the more challenging skills that our pupils learn at GCSE. I have seen many pupils struggle with it, even those who achieve a grade 8 or 9 at GCSE and go onto take maths at A level. Interestingly, it doesn't seem like it should be challenging at all. I think it's way more straightforward than some of the harder questions that come up at GCSE - so why do pupils struggle with it so much?
When I first became a teacher, I taught my pupils to factorise harder quadratics in the way I've always done it: by inspection. Simply write out two empty pairs of brackets and try some numbers - thinking logically about what those numbers could be - until your terms expand correctly. It's very quick once you get the hang of it. In my NQT year this method was met with frustration by my Year 11s. They wanted a more defined procedure. I looked online to see if I was missing something and discovered 'the grouping method' which I then showed them as an alternative. I felt that it was an unnecessarily convoluted method but they clearly preferred having a set of rules to follow rather than having to reason for themselves. It made me a bit sad.
This grouping method, particularly the last step where terms are gathered together, is a bit of a leap of faith for pupils who have never seen this kind of factorisation before. It kind of seems like magic.
Ten years on, I still prefer inspection (the 'guess and check' method) but I always teach my pupils the grouping method as an alternative. I know they'll see it elsewhere even if I don't teach it to them - in textbooks, revision guides and online videos. I find that only the strongest pupils favour inspection - most pupils choose to use the grouping method but very often forget it. Days before the GCSE exam I hear cries of 'what's that thing you do to factorise hard quadratics? Something to do with the middle term...?'. The steps in the grouping method are not intuitive, and as a result it's difficult to remember.
Given I had never heard of the grouping method before I started teaching, I was surprised to learn that it's actually an incredibly popular method. In fact, it appears to be the method that most maths teachers now use to teach non-monic factorisation.
How did I miss this method during my time at school? I did my maths GCSE in 90s. I have a couple of Bostock and Chandler textbooks from the 1990 - both only feature inspection, with no mention of grouping.
This probably explains why I'd not seen it before. It wasn't in fashion when I was at school.
Looking further back I was surprised to see that older textbooks do feature the grouping method. Here, in New Algebra for Schools (Durell, 1953), we see an example of the grouping method. Durell recommends this method for both monic and non-monic quadratics.
Note though that this follows on from extensive use of grouping elsewhere. By this point in the textbook pupils have had considerable experience of factorising expressions like p(a + b) + q(a+b) and ax - ay + bx - by. This is absolutely key. There is a clear progression here that I feel is often missing from modern teaching of factorisation. I'm not sure it make much sense for pupils to only use the grouping method for non-monic quadratics, having never seen it before.
Later, this chapter tells us that simple quadratic functions can often be factorised at sight without using the grouping method. It says "use the grouping method whenever you are not able to obtain the factors by inspection, quickly".
Another textbook from eight years later (The Essentials of School Algebra, Mayne, 1961) also features both the grouping and inspection methods. Again, earlier in the chapter there is a considerable amount of work on the skills and understanding required for the grouping method. Of inspection it says,
"After a little practice, the pupil will be able to reject mentally the impossible pairs of factors... With simple numbers it is slightly quicker than the method of grouping terms, but the grouping method is the method to rely on. It should always be used whenever the pupil is not able to obtain the factors quickly be inspection."
So it seems that the grouping method may have been popular in the mid-20th Century.
Looking even further back, in Elementary Algebra for Schools (Hall & Knight, 1885) we are told that we should factorise non-monics by inspection:
"The beginner will frequently find that is it not easy to select the proper factors at the first trial. Practice alone will enable him to detect at a glance whether any pair he has chosen will combine so as to give the correct coefficients of the expression to be resolved".
There's no mention of any alternative methods here.
I haven't yet read enough old textbooks to track the full history of the grouping method, but from what I've seen it has come and gone over the years, and sadly seems to have become detached from prerequisite skills along the way.
If teachers are teaching the grouping method to factorise non-monics and have not already taught pupils how to factorise expressions like p(a + b) + q(a + b) and ax - ay + bx - by then I think they may be trying to teach too many new skills in one go. It's certainly something to think about.
If you find this kind of thing interesting, come along to my workshop at MathsConf17. I start will by sharing some cool stuff I've seen in old maths textbooks, and then I will focus on quadratics. We won't look at factorising, but we will look at some methods for solving quadratic equations and how they've changed over the years. I really think that this kind of subject knowledge development helps to improve our classroom practice. I hope to see you there - and I will bring along some of my old textbooks in case you'd like to have a look for yourself.