Teaching unit · UK Year 9 (Key Stage 3, ages 13 to 14)

Algebraic manipulation: expanding double brackets and rearranging formulae

Expanding the product of 2 binomials, and rearranging a standard formula to make a different letter the subject

About three lessons of 45 to 60 minutes

Start here · hook

2 different algebra skills, 1 shared goal: an equivalent, more useful expression

Expanding double brackets and rearranging formulae look different, but they share a purpose: rewriting an expression or equation into a more USEFUL equivalent form, without changing what it actually means. (x + 3)(x + 5) and $x^{2}$ + 8x + 15 are the exact same expression, just written differently, and one of them is far easier to substitute a number into. Likewise, v = u + at and t = (v - u)/a describe the exact same relationship between the 4 quantities, just solved for a different letter.

Both skills rely on the same underlying idea, whatever you do to one side of an equals sign (or one part of an expression), you must do consistently, so the equivalence is never broken.

(x + 2)(x + 3) expanded is $x^{2}$ + 5x + 6the same value for any x, just multiplied out
Area = length x width, rearranged for widthwidth = Area / length, the same fact, solved for a different letter
F = 9C/5 + 32, rearranged for Clets you convert Fahrenheit back to Celsius using the same formula
v = u + at, rearranged for tfinds how long acceleration took, from the starting and final speeds

Learning objective

What students will be able to do

Students will expand the product of 2 binomials (including the difference-of-squares special case) by multiplying every term in the first bracket by every term in the second, and rearrange standard formulae to make a different letter the subject, using the same balancing operations as solving an equation.

Success criteria

I can expand (x + a)(x + b) by multiplying every term in the first bracket by every term in the second.
I can expand brackets with a coefficient greater than 1 in front of x, such as (2x + a)(3x + b).
I can recognise and expand the difference-of-squares pattern (x - a)(x + a).
I can rearrange a formula to make a different letter the subject, undoing operations in reverse order.

Curriculum anchor

Standards this unit teaches

KS3 Maths: AlgebraUK National Curriculum (England)
Algebra
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Algebra" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "simplify and manipulate algebraic expressions to maintain equivalence by:...expanding products of 2 or more binomials"; "understand and use standard mathematical formulae; rearrange formulae to change the subject".

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Binomial: an expression with exactly 2 terms, such as x + 3 or 2x - 5
Expand: to multiply out brackets, removing them while keeping the expression equivalent
Like terms: terms with exactly the same letter part (such as 2 different x terms), which can be combined by adding or subtracting
Difference of squares: the pattern (x - a)(x + a) = $x^{2}$ - $a^{2}$ , where the middle terms always cancel exactly
Subject of a formula: the letter on its own on 1 side of a formula, such as v in v = u + at
Rearrange (change the subject): to use inverse operations to make a DIFFERENT letter become the subject of a formula

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Expanding double brackets: the area-model connection

Concrete

Before expanding algebra brackets, look at the same idea with plain numbers. 13 x 12 can be split as (10 + 3)(10 + 2), and multiplying that out area-model style gives 4 separate products, 10x10, 10x2, 3x10 and 3x2, which add up to the total. Expanding (x + 3)(x + 2) works EXACTLY the same way, just with x standing in for the '10' part.

Multiplying 2 binomials always produces exactly 4 products (sometimes called FOIL: First, Outer, Inner, Last), and the 2 'middle' products (Outer and Inner) are usually like terms that combine into a single x term.

13 x 12 split as (10 + 3)(10 + 2): the 4 regions are 10x10 = 100, 10x2 = 20, 3x10 = 30 and 3x2 = 6, adding to 156. Expanding (x + 3)(x + 2) follows the exact same 4-region pattern, with x in place of 10.

Worked example

Expand and simplify (x + 3)(x + 2).

Multiply every term in the first bracket by every term in the second: x times x = $x^{2}$ , x times 2 = 2x, 3 times x = 3x, 3 times 2 = 6.
Combine the 2 like x terms: 2x + 3x = 5x.

Answer: $x^{2}$ + 5x + 6.

Check for understanding, ask

Why does multiplying 2 binomials always give exactly 4 products before simplifying?
Expand (x + 1)(x + 4).

2. Negative terms and a coefficient greater than 1

Abstract

The same 4-product method works with negative numbers and with a coefficient greater than 1 in front of x, just track the signs carefully term by term. A negative times a negative is positive, and a positive times a negative is negative, exactly as in ordinary number arithmetic.

Worked example

Expand and simplify (x - 4)(x - 3), and (2x + 1)(3x - 5).

(x - 4)(x - 3): x times x = $x^{2}$ , x times -3 = -3x, -4 times x = -4x, -4 times -3 = 12. Combine: -3x - 4x = -7x.
(2x + 1)(3x - 5): (2x)(3x) = 6x^2, (2x)(-5) = -10x, (1)(3x) = 3x, (1)(-5) = -5. Combine: -10x + 3x = -7x.

Answer: (x - 4)(x - 3) = $x^{2}$ - 7x + 12. (2x + 1)(3x - 5) = 6x^2 - 7x - 5.

Check for understanding, ask

In (2x + 1)(3x - 5), why is the $x^{2}$ coefficient 6, not 5?

3. The difference-of-squares pattern

Abstract

When the 2 brackets contain the same number, once with a plus and once with a minus, like (x - 5)(x + 5), the 2 middle terms are always OPPOSITES, so they cancel exactly, leaving only $x^{2}$ minus the number squared. Spotting this pattern saves time and avoids an unnecessary step.

Worked example

Expand and simplify (x - 6)(x + 6).

x times x = $x^{2}$ , x times 6 = 6x, -6 times x = -6x, -6 times 6 = -36.
The middle terms 6x and -6x cancel exactly (add to 0).

Answer: $x^{2}$ - 36.

Check for understanding, ask

Predict the expansion of (x - 9)(x + 9) without multiplying it out term by term.

4. Rearranging formulae to change the subject

Abstract

A formula like A = lw connects 3 (or more) quantities. Whichever letter is alone on 1 side is called the SUBJECT. To make a DIFFERENT letter the subject, use the same balancing moves as solving an equation, undo the operations connecting the letters, in reverse order, doing the same thing to both sides every time.

A useful check afterward: substitute real numbers into the ORIGINAL formula to get a value, then substitute the SAME numbers into the rearranged version and confirm it gives back the expected answer.

Worked example

Make w the subject of A = lw. Make t the subject of v = u + at.

A = lw: divide both sides by l: A / l = w, so w = A / l.
v = u + at: subtract u from both sides: v - u = at. Divide both sides by a: t = (v - u) / a.

Answer: w = A / l. t = (v - u) / a.

Check for understanding, ask

Why must the SAME operation be applied to both sides of a formula when rearranging it?
Make l the subject of P = 2l + 2w.

Watch for

Common misconceptions and how to address them

Misconception(x + a)(x + b) expands to just $x^{2}$ + ab (multiplying the whole brackets together like 2 single numbers).

Why it happens: Students over-generalise the single-number multiplication rule and skip the requirement to multiply EVERY term by EVERY term.

How to address it: Use the area-model rectangle (4 distinct regions) every time until the 4-product habit is automatic; count out loud '1, 2, 3, 4 products' before combining.

MisconceptionIn (x - 4)(x - 3), the constant term is found by adding 4 and 3 (getting +7), instead of multiplying negative 4 by negative 3.

Why it happens: Confusing the RULE for the x-coefficient (which comes from adding the 2 numbers) with the rule for the constant term (which comes from MULTIPLYING them).

How to address it: Keep the 2 rules visually separate: x-coefficient = sum of the 2 numbers, constant term = PRODUCT of the 2 numbers. Verify with a fully worked-out 4-term expansion each time.

MisconceptionWhen rearranging v = u + at to find u, you divide by a and t instead of just subtracting u's neighbour term.

Why it happens: Students apply a 'reverse everything' instinct without checking which operation (addition, not multiplication) actually connects u to the rest of the equation.

How to address it: Identify the LAST operation applied to the subject in the original formula, and undo operations in the OPPOSITE order they were applied (last operation undone first).

Do it together

Guided practice (with answers)

1. Expand and simplify (x + 5)(x + 2).
Answer: $x^{2}$ + 7x + 10, from $x^{2}$ + 2x + 5x + 10.
2. Expand and simplify (x - 3)(x - 8).
Answer: $x^{2}$ - 11x + 24, from $x^{2}$ - 8x - 3x + 24.
3. Expand and simplify (x - 5)(x + 5).
Answer: $x^{2}$ - 25 (difference of squares: the middle terms cancel).
4. Expand and simplify (2x + 3)(x - 1).
Answer: 2x^2 + x - 3, from 2x^2 - 2x + 3x - 3.
5. Make b the subject of P = 2a + 2b.
Answer: b = (P - 2a) / 2, by subtracting 2a then dividing by 2.
6. Make R the subject of I = PRT / 100.
Answer: R = 100I / (PT), by multiplying both sides by 100 then dividing by PT.

On their own

Independent practice worksheets

Practise expanding double brackets and rearranging formulae with computed, never-wrong answer keys.

Year 9 expanding double brackets worksheetthe core skill from sections 1-3 Year 9 rearranging formulae worksheetthe change-the-subject skill from section 4

Reach every student

Differentiation

Support

For double brackets, draw the area-model rectangle every time until the 4-product habit is automatic, even for symbolic (non-numeric) brackets.
Colour-code the 4 products (e.g. First=red, Outer=blue, Inner=green, Last=purple) to make the FOIL order visible on the page.
For rearranging formulae, write out the formula's 'flow diagram' (what operation happens to the subject, in order) before reversing it.
Practise rearranging 1-step formulae (like A = lw) fluently before tackling 2-step ones (like v = u + at).

Extension

Expand a product of 3 binomials, e.g. (x + 1)(x + 2)(x + 3), by expanding 2 first and then multiplying the result by the 3rd.
Explain algebraically why (x + a)(x - a) always equals $x^{2}$ - $a^{2}$ for ANY value of a, using the general expansion.
Rearrange a formula where the target letter appears in 2 different terms (e.g. make x the subject of ax + b = cx + d).
Investigate whether rearranging a formula changes its value for a given set of numbers (it should not: verify with a numeric substitution check).

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling double-bracket expansion (including difference of squares) and rearranging.

1. Expand and simplify (x + 6)(x - 2).
Answer: $x^{2}$ + 4x - 12, from $x^{2}$ - 2x + 6x - 12.
2. Expand and simplify (x - 7)(x + 7).
Answer: $x^{2}$ - 49 (difference of squares).
3. Make C the subject of F = 9C/5 + 32.
Answer: C = 5(F - 32) / 9, by subtracting 32 then multiplying by 5 then dividing by 9.

For the teacher

Teacher notes and timings

Rough timing across 3 lessons: Lesson 1 area-model and basic double brackets (section 1), Lesson 2 negatives/coefficients and difference of squares (sections 2-3), Lesson 3 rearranging formulae (section 4) plus the exit ticket.
The area-model figure in section 1 uses a NUMERIC example (13 x 12), reusing the site's existing array/split figure engine exactly as the site's whole-number multiplication units do; no new figure kind was added. The connection to algebra (x standing in for 10) is made explicitly in the caption and body text, since the array figure itself only draws whole-number cell counts, not symbolic x's.
Every rearranged formula in the matching worksheet comes from a fixed, hand-verified pool of 6 standard formulae (SUVAT, rectangle area, perimeter, simple interest, Fahrenheit-Celsius, Ohm's law); the test suite verifies every rearrangement algebraically by substituting random numbers and checking the forward and rearranged formulae agree, not just by re-typing the same answer twice.

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