*************
Factorising
a quadratic when the coefficient of x2 doesn’t equal one (a
‘non-monic’) is apparently one of the more challenging skills that our pupils learn
at GCSE. I’ve 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 reasoning 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.
 |
| The grouping method, explained in a GCSE maths textbook published by OUP in 2016 |
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 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.
 |
| Grouping to factorise quadratics in the 1950s |
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.
 |
| The inspection method in use in the 1990s |
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 done any kind of factorising by grouping before.
 |
| An exercise in factorising by grouping, to be completed prior to learning how to factorise quadratics by grouping |
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 something to
think about.
********
