S2 Maths | Identities and Factorisation Lesson-Prep Guide (for tutors) | 60 minutes from expansion to cross-multiplication

2026-11-07

Tutor lesson-prep guides · Secondary Maths

Most students can expand but freeze when asked to factorise, because expanding runs forwards while factorising means guessing backwards. The lesson is not about memorising tricks: it is about turning expansion into a reversible move.

Objectives — Students can: (1) state and apply the three identities (a+b) squared, (a-b) squared and a squared minus b squared without multiplying term by term; (2) take out the common factor first, then factorise x squared plus bx plus c by cross-multiplication; (3) decide within 30 seconds which method a polynomial needs and check the answer by expanding it back.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Distributive law	Expand 3(x+4)	3x+12, multiply each term
Multiplying negatives	What is (-2) x (-3)?	+6, same signs give plus
Like terms	Can 5x and 3x squared be added?	No, the powers differ
Squaring a term	Is (2x) squared equal to 2x squared or 4x squared?	4x squared, the coefficient is squared too

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-15 min	Opening contrast	Write (x+3)(x+4) on the left of the board and x squared plus 7x plus 12 on the right; ask what the link is	Say the two forms are the same expression
15-28 min	The three identities	Show (a+b) squared, (a-b) squared and a squared minus b squared, using an area diagram to explain the 2ab middle term	Write the three forms and do one question each
28-38 min	Common factor first	Show 6x squared plus 9x = 3x(2x+3): numeric HCF first, then the lowest power of each letter	Five questions on taking common factors
38-50 min	Cross-multiplication	Show x squared plus 7x plus 12: find two numbers multiplying to 12 and adding to 7, write (x+3)(x+4), expand to check	Six questions, each expanded back as a check
50-57 min	Negative cases	Show x squared minus 5x minus 24: the product is negative, the sum negative, so one number is negative and the larger one positive	Three questions with negative signs
57-60 min	Exit check	Three quick oral questions	Independent answers

03 Teaching script (read it out)

Scene 1: Factorising is expansion played backwards, not a new formula

Students treat factorising as a pile of techniques, so every question starts with “which method is this one”. In fact there is one goal: rewrite a sum as a product. Every method serves that goal, and the check is always the same, expand the answer and compare with the original.
Say this in the first minute: expanding goes from brackets to a polynomial, factorising goes from a polynomial back to brackets. Once students hold that picture, the methods are just tools.

Students do: explain the difference between expanding and factorising in one sentence, with one example of each.

Scene 2: Cross-multiplication is a search for two numbers, not a cross shape

Students draw tidy crosses and still get it wrong, because they never asked which two numbers multiply to the constant and add to the coefficient of x. It is a factor-pair search: for 12 the pairs are (1,12), (2,6) and (3,4), and only (3,4) adds to 7.
Make them list every pair, add each one, circle the winner, and only then write the brackets. It feels slower, but it explains why the answer is unique.

Students do: list all three factor pairs of 12 with their sums before factorising x squared plus 7x plus 12.

Scene 3: The common factor comes first, every time

Students see 3x squared plus 6x and reach straight for cross-multiplication, then get stuck because 3 and x are common to both terms. The order is fixed: numeric HCF, then the lowest power of each letter, then cross-multiplication or an identity on what remains.
Taking the common factor also prevents missing answers: quite often what is left inside the bracket is a difference of squares that is obvious at a glance. Train the habit of writing a small “CF” beside each question before starting.

Students do: before each question, state whether there is a common factor and what it is.

04 Question types and worked steps

Type	Example	Steps	Answer
Direct expansion with an identity	(2x-5) squared	Recognise (a-b) squared, set a = 2x and b = 5, middle term 2ab = 20x	4x squared minus 20x plus 25, the middle term must be there
Difference of squares	(3a+2)(3a-2)	Same a, one plus and one minus, so the answer has only two terms	9a squared minus 4, the middle terms cancel
Common factor	6x squared plus 9x	HCF of 6 and 9 is 3, lowest power of x is x	3x(2x+3)
Cross-multiplication with negatives	x squared minus 5x minus 24	List pairs of 24, find the pair summing to -5: -8 and +3	(x-8)(x+3), check by expanding

Sanity check (teach this): Sanity checks: expand every factorised answer and compare both the x term and the constant; a difference of squares must have exactly two terms, so any middle term means an error; the middle term of a perfect square is 2ab and students routinely drop the 2; nothing inside the bracket may still carry a common factor; for cross-multiplication the product of the two numbers must equal the constant including its sign, and the sum must equal the coefficient of x.

05 Independent practice (3 questions)

Question	Answer	What to watch
Five common-factor questions	Write the common factor before factorising	Check whether the letter part is dropped
Six cross-multiplication questions	List every factor pair of the constant first	Check that pairs are complete and signs are right
Three mixed questions	No labels, student chooses the method	Check whether the method is chosen within 30 seconds

06 Common mistakes and what to say

Mistake	Student writes	Say this
Writing (2x) squared as 2x squared	First term of (2x-5) squared typed as 2x squared	Both the 2 and the x are squared, so the first term is 4x squared.
Dropping the middle term	(x+4) squared answered as x squared plus 16	A perfect square has three terms; the middle one is 8x and leaving it out breaks the identity.
Giving three terms for a difference of squares	(x+3)(x-3) answered as x squared plus 9	One plus, one minus: the middle terms cancel, so only two terms survive.
Not checking the answer	x squared minus 5x minus 24 answered as (x-8)(x-3)	Expand it: the middle term is minus 11x, not minus 5x. Thirty seconds of checking saves five minutes of redoing.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters for this subject
Lesson-prep guide index — Matching chapters in other subjects
Redo this chapter — Redo the six cross-multiplication questions with factor pairs and checks

08 Exit check (last 5 minutes)

Question	Answer	Decision
Expanding versus factorising	One removes brackets, the other restores them	Answering “same thing” means returning to act one
How many terms in (a-b) squared	Three, with minus 2ab in the middle	Answering two means returning to act two
First step in any factorisation	Ask whether there is a common factor	Going straight to cross-multiplication means returning to act three
The two numbers for x squared minus 5x minus 24	-8 and +3	Wrong signs mean returning to act five

09 Tutor FAQ

What if the student is slow at times tables?

Start with constants such as 12 or 24 that have several pairs and require the pair table; speed follows familiarity with the pairs.

Memorise the identities or expand each time?

The three identities are worth memorising because those questions rely on recognition; everything else is checked by expansion.

When should a cubed identity be introduced?

Not in S2: keep the quadratic identities and cross-multiplication clean to avoid mixing rules.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.