Factorising quadratics

Many students who are comfortable factorising quadratics of the form x2 + bx + c struggle when there is a coefficient in front of the x term, as in ax2 + bx + c. They usually understand what is required, but become frustrated through the apparent ‘trial and error’ process that they apply until they identify the correct solution.

When I was a student at school, I remember being taught a structured trial and error process that worked methodically through all the possible combinations. It worked every time, but it was laborious and the greater the number of factors that the ‘a’ and ‘c’ terms had, the more tiresome the process became.

As a teacher, I continually look for improved methods of carrying out this process. However, many of them lose the understanding of what is going on – and students can end up struggling to remember a method, rather than remember what they are trying to achieve. To this end, I prefer for students to begin with a trial and error method, so that they appreciate the nature of the problem, before moving onto an appropriate method.

Consider the example 2x2 – x – 6 in order to study some of those methods.

Most methods start by multiplying the ‘a’ and ‘c’ terms…

…then finding factor pairs of their product, i.e. of -12…

…and identifying the pair that sums to give the ‘b’ term (i.e. -1). In the chosen example, this gives 3 and -4.

The ‘Lizzie method’ separates the pair, as shown, divides by the ‘a’ term, then multiplies the ‘x’ and the fraction (simplifying if necessary) until the denominator has been multiplied out.

A variation (see here for more detail) on the Lizzie method, separates the pair and multiplies the ‘x’ by the ‘a’ term, dividing by any common factors.

However, in my mind, these are rather formulaic and I prefer a method which is more intuitive. The method that I like to use (see here for more detail) involves rewriting the ‘b’ term as the sum of the factor pairs. The expression is then factorised in two parts; there is no need to have to ‘remember’ which number to write where, and which terms to multiply or divide by other terms.

OrHere is another example, 6x2 – 5x – 6.

Multiply ‘a’ and ‘c’ terms  Find factor pairs of -36  First method
 Second method  Third method  Third method, again

However, as a teacher, judgement is required as to which method, if any, may be more appropriate for the class, or just for individuals within it, to use. Experience has shown that time spent explaining lots of methods to lots of students can be detrimental. Many students don’t want to see more than one method, so finding time (*sigh*) to individualise explanations would be beneficial – even if this is done ‘out of class’, or otherwise.

Beast Beyond Belief

One student in class today was factorising quadratic equations and was struggling, so I showed him the ‘Lizzie’ method http://tinyurl.com/2u6funv. He is one of the cleverer students and appreciates that there are often different ways to solve problems. He could also see that I was largely working it out as I was going along (well, I did tell him that this was the case), but as we approached the answer, he proclaimed:

“sir, you are a beast beyond belief”

I smiled to myself. I had managed to ‘do it’ and help him but, also, I had received praise from a student. It doesn’t happen very often, but reminds me of my belief that it is not the nature of the praise that is important – more the fact that there is an acknowledgement of ‘doing good’.

Many argue that some students don’t appreciate praise. However, I believe that while it may not be cool to receive praise, particularly in front of peers, the recognition (and confirmation) is hugely appreciated – even if only in secret. I also don’t believe the ‘size’ of the praise matters. While a monetary reward acts as a carrot to pull, in the same way that a prison sentence acts as a stick to push, a simple ‘merit’ or ‘demerit’ is enough to provide confirmation of good or bad. As such, my lessons tend to be full of lots of relatively meaningless (but well intentioned) positives (credits/good comments/commendations/merits etc.) and negatives (verbal and visual warnings).

My success rate is determined by my consistency of application.