Factorising is pattern-spotting, and patterns can be drilled

Factorising is the topic where confident students suddenly go quiet. They can expand brackets all day — that direction feels mechanical. Going the other way feels like magic, like you're supposed to just see the answer, and if you don't, you're stuck.

But there's no magic. Factorising is recognising which of a small number of patterns you're looking at, and pattern recognition is a trainable skill, not a talent you're born with.

There are only a few shapes

Nearly everything at this level is one of these, and the whole game is identifying which:

• Common factor. Every term shares something. $6x^2 + 9x = 3x(2x + 3)$. Always check this first — it's the most-missed mark in the topic.
• Difference of two squares. $a^2 - b^2 = (a+b)(a-b)$. The tell is a subtraction of two perfect squares and no middle term. $x^2 - 25 = (x+5)(x-5)$.
• Quadratic trinomial. $x^2 + 5x + 6$. Find two numbers that multiply to give the last term and add to give the middle one. Here, $2$ and $3$: $(x+2)(x+3)$.

That's most of GCSE-level factorising in three bullet points. The student who "can't do it" almost always just hasn't been told it's a menu, not an open field.

Train the eye, not the answer

You don't get faster at factorising by understanding it more deeply — you understand it fine. You get faster by seeing the shape sooner. So I drill recognition directly:

1. Sort, don't solve. Give a student twenty expressions and ask only "which pattern is each?" — no factorising yet. Naming the shape is the hard part; the rest is routine.
2. Always strip the common factor first. $2x^2 - 8 = 2(x^2 - 4) = 2(x+2)(x-2)$. Skip step one and the difference of squares hides.
3. Expand to check, every time. Factorising has a built-in answer key — multiply your brackets back out and see if you land where you started.

You're not searching a blank space. You're matching against a short list of shapes you already know. The skill is recognising the shape fast, and that only comes from reps.

A student who's done forty of these stops "trying to factorise" and starts seeing "oh — difference of two squares" on sight. That shift, from solving to recognising, is the whole thing.