What is algebraic notation and why is it important in AQA GCSE Mathematics?

Algebraic notation is a system of symbols and rules used to represent mathematical concepts and relationships. In AQA GCSE Mathematics, it is crucial because it allows students to express complex ideas succinctly and solve equations systematically. Understanding algebraic notation helps in manipulating expressions and solving problems efficiently.

How do you simplify algebraic expressions in GCSE Mathematics?

To simplify algebraic expressions, you combine like terms, which are terms that have the same variable raised to the same power. For example, in the expression 3x + 5x, you can combine the terms to get 8x. Simplifying expressions is a key skill in AQA GCSE Mathematics as it makes solving equations more manageable.

What is the difference between a term and a coefficient in algebra?

In algebra, a term is a single mathematical expression that can be a number, a variable, or a combination of both, such as 5, x, or 3x. A coefficient is the numerical part of a term that contains a variable. For example, in the term 3x, 3 is the coefficient. Understanding these concepts is essential for manipulating algebraic expressions in AQA GCSE Mathematics.

How do you expand brackets in algebraic expressions?

To expand brackets in algebraic expressions, you multiply each term inside the bracket by the term outside. For example, to expand 2(x + 3), you multiply 2 by x and 2 by 3, resulting in 2x + 6. Mastering this technique is important for solving more complex problems in AQA GCSE Mathematics.

What are common mistakes to avoid when manipulating algebraic expressions?

Common mistakes include failing to combine like terms correctly, misapplying the distributive property when expanding brackets, and neglecting to apply the correct order of operations. In AQA GCSE Mathematics, careful attention to these details is crucial for accurate manipulation and simplification of algebraic expressions.

Area model expansion (x + 3)(x + 2) = x² + 5x + 6

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At a glance

Algebraic notation is shorthand: $ab = a \times b$ , $a^2 = a \times a$ , $\dfrac{a}{b} = a \div b$
Key vocabulary: term, expression, equation, formula, identity, factor, coefficient
An identity ( $\equiv$ ) is true for all values; an equation is true for specific values
Substitution replaces letters with numbers — always bracket negative values
Like terms share the same letter(s) and power(s) and can be collected
Expanding removes brackets; factorising reverses the process
To rearrange a formula, apply inverse operations; if the subject appears twice, collect and factorise

What you should be able to do

Use and interpret algebraic notation including $ab$ , $3y$ , $a^2$ , $a^n$ , $\frac{a}{b}$ and brackets (A1)
Substitute numerical values into formulae and expressions, including scientific formulae (A2)
Understand and use the vocabulary: expression, equation, inequality, term, factor, identity (A3)
Simplify by collecting like terms, expanding brackets, and taking out common factors (A4)
Expand products of two or more binomials and factorise quadratic expressions, including $ax^2 + bx + c$ (A4)
Rearrange formulae to change the subject, including cases where the subject appears more than once (A5)
Distinguish between an equation and an identity; construct and interpret algebraic arguments (A6)

Study notes

Algebraic Notation

The shorthand rules for writing algebra — AQA spec A1

Algebra uses letters to represent unknown or general numbers. Specific notation rules allow us to write expressions compactly and unambiguously.

Multiplication — the × sign is omitted:

$a \times b$ is written $ab$
$3 \times y$ is written $3y$ (coefficient always comes first)
$3 \times (x + 2)$ is written $3(x + 2)$

Repeated multiplication — powers (indices):

$a \times a = a^2$
$a \times a \times a = a^3$
$a$ multiplied by itself $n$ times $= a^n$

Division — written as a fraction: $a \div b = \dfrac{a}{b}$

Coefficients are fractions, not decimals: $0.5x \text{ should be written } \dfrac{x}{2}$

$5 \times p \times q = 5pq$
$x \times x \times x \times x = x^4$
$2 \div y = \dfrac{2}{y}$
$-x$ has coefficient $-1$ ; $\dfrac{x}{3}$ has coefficient $\dfrac{1}{3}$

Common pitfall

Never write $3y$ as $y3$ — the coefficient always comes before the letter. Similarly, $3x^2$ means $3 \times (x^2)$ , not $(3x)^2$ .

Key Vocabulary

Expressions, equations, identities, formulae — AQA spec A3, A6

Precise vocabulary is tested directly in AQA 8300 exam questions.

Name	Meaning	Example
Term	A single number, letter, or product	$3x$ , $-2y^2$ , $7$
Expression	Terms combined with $+$ or $-$ ; no $=$ sign	$3x^2 - 2x + 1$
Equation	Two expressions set equal; true for specific values	$2x + 1 = 7$
Formula	A rule connecting two or more variables	$v = u + at$
Identity	True for all values; written with $\equiv$	$3(x+2) \equiv 3x+6$
Inequality	Shows one side is greater or less than another	$2x + 1 > 5$
Factor	Divides exactly into another expression	$3$ and $x$ are factors of $3x$

Equation vs Identity (A6):

$2x + 1 = 9$ is an equation — true only when $x = 4$ .

$2(x + 1) \equiv 2x + 2$ is an identity — true for every value of $x$ .

Common pitfall

Do not confuse an equation with an identity. Exam questions that say 'prove' or 'show this is an identity' require algebraic working — a single numerical example is never sufficient.

Substitution

Evaluating expressions by replacing letters with numbers — AQA spec A2

Substitution means replacing every letter with a given number, then evaluating using BIDMAS.

Method:

Write out the expression.
Replace each letter with its value, using brackets.
Evaluate carefully, applying BIDMAS.

Example 1: Find $3a^2 - 2b$ when $a = -2$ , $b = 5$ :

$3(-2)^2 - 2(5) = 3(4) - 10 = 12 - 10 = 2$

Example 2 — scientific formula: Kinetic energy $E_k = \dfrac{1}{2}mv^2$ .

Find $E_k$ when $m = 4\text{ kg}$ , $v = 3\text{ m/s}$ :

$E_k = \frac{1}{2}(4)(3)^2 = \frac{1}{2}(4)(9) = 18 \text{ J}$

Always bracket a negative substitution: $(-2)^2 = 4$ , not $-4$
$3a^2$ means $3 \times (a^2)$ — square $a$ first, then multiply by 3
Check units when substituting into scientific formulae

Common pitfall

$(-3)^2 = 9$ but $-3^2 = -9$ . The bracket changes everything — always include it when substituting negative values into a power.

Collecting Like Terms

Simplifying by adding and subtracting terms — AQA spec A4

Like terms have exactly the same letter part (including powers). Only like terms can be added or subtracted.

Pair	Like?	Reason
$3x$ and $5x$	✓	Same letter, same power
$4x^2$ and $-x^2$	✓	Both $x^2$
$2xy$ and $5xy$	✓	Both $xy$
$3x$ and $3x^2$	✗	Different powers
$5x$ and $5y$	✗	Different letters

Example: Simplify $4x^2 + 3x - 2x^2 + x - 5$ :

$(4x^2 - 2x^2) + (3x + x) - 5 = 2x^2 + 4x - 5$

$xy$ and $yx$ are like terms — multiplication is commutative
Constants (plain numbers) are like terms with each other
In $3a^2b$ , both the letters and the powers must match for terms to be like

Expanding Brackets

Single brackets, double brackets, and special products — AQA spec A4

Single bracket: Multiply every term inside by the term outside.

$a(b + c) = ab + ac$

Example: Expand $3x(2x - 5) = 6x^2 - 15x$

Double brackets — FOIL (First, Outer, Inner, Last):

$(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$

Example: Expand $(2x + 1)(3x - 4)$ :

$= 6x^2 - 8x + 3x - 4 = 6x^2 - 5x - 4$

Special products to memorise:

$(a + b)^2 = a^2 + 2ab + b^2$ $(a - b)^2 = a^2 - 2ab + b^2$ $(a + b)(a - b) = a^2 - b^2 \quad \text{(difference of two squares)}$

Area model for (x + 3)(x + 2): the rectangle splits into four cells whose areas sum to x² + 5x + 6.

Expand $-(x - 3)$ as $-x + 3$ , not $-x - 3$
$(x + 3)^2 \neq x^2 + 9$ — the middle term $2(x)(3) = 6x$ must be included
Always collect like terms after expanding

Common pitfall

$(x + 3)^2 \neq x^2 + 9$ . Expand as $(x+3)(x+3)$ to get $x^2 + 6x + 9$ . Missing the $2ab$ middle term is the most common error in GCSE algebra.

Factorising

Common factor, quadratics, and difference of two squares — AQA spec A4

Factorising rewrites an expression as a product of factors — it is the reverse of expanding.

1. Highest common factor (HCF):

$6x^2 + 9x = 3x(2x + 3)$

2. Quadratic $x^2 + bx + c$ :

Find two numbers that multiply to $c$ and add to $b$ .

$x^2 + 5x + 6 = (x + 2)(x + 3) \quad (2 \times 3 = 6,\ 2 + 3 = 5)$

3. Quadratic $ax^2 + bx + c$ (Higher — AC method):

Find two numbers that multiply to $ac$ and add to $b$ , then split the middle term.

$2x^2 + 7x + 3: \quad ac = 6,\ 1+6=7$ $= 2x^2 + x + 6x + 3 = x(2x+1) + 3(2x+1) = (x+3)(2x+1)$

4. Difference of two squares:

$a^2 - b^2 = (a + b)(a - b)$

$9x^2 - 25 = (3x + 5)(3x - 5)$

Reverse area model: the cells of x² + 7x + 10 reassemble into a rectangle of (x + 2) by (x + 5).

Always look for a common factor first before trying other methods
Verify by expanding your answer — you should recover the original expression
$x^2 + 9$ cannot be factorised — only a difference (not a sum) of squares factorises

Common pitfall

$x^2 - 9 = (x+3)(x-3)$ (difference of two squares), but $x^2 + 9$ does not factorise over the real numbers. A sum of two squares has no real factors.

Rearranging Formulae

Changing the subject, including cases where the subject appears twice — AQA spec A5

Rearranging a formula means isolating a chosen variable (the new subject) on one side using inverse operations.

Basic rearrangement:

Make $u$ the subject of $v = u + at$ :

$u = v - at$

Make $r$ the subject of $A = \pi r^2$ :

$r^2 = \frac{A}{\pi} \implies r = \sqrt{\frac{A}{\pi}}$

Subject appears twice (Higher):

Collect all terms containing the subject on one side, then factorise before dividing.

Make $x$ the subject of $\dfrac{x}{x+2} = y$ :

$x = y(x + 2)$ $x = yx + 2y$ $x - yx = 2y$ $x(1 - y) = 2y$ $x = \dfrac{2y}{1 - y}$

If the target variable is in a denominator, multiply both sides first to clear it
If the target variable appears on both sides, collect those terms together, then factorise
When square-rooting, consider whether $\pm$ is appropriate in context

Common pitfall

When the subject appears twice, you must collect all terms containing it on one side and then factorise before dividing. Dividing too early leaves the variable in two places and gives an incomplete answer.

Quick recap

Notation: $ab = a \times b$ , $a^2 = a \times a$ , $\frac{a}{b} = a \div b$ ; coefficients are fractions not decimals
Expression — no equals sign; Equation — true for specific values; Identity ( $\equiv$ ) — true for all values; Formula — connects variables
Substitution: replace letters with bracketed numbers, then evaluate with BIDMAS
Like terms share the same letter(s) and power(s) — only these can be collected
Expanding: multiply through the bracket; use FOIL for double brackets; $(a+b)^2 = a^2 + 2ab + b^2$
Factorising: HCF first; then find two numbers (× $c$ , + $b$ ) for quadratics; $(a+b)(a-b)$ for difference of two squares
Rearranging: use inverse operations; if the subject appears twice, collect all those terms and factorise before dividing

Exam tips

Show all steps when rearranging a formula — method marks are awarded for clear algebraic working
To prove an identity, start from the more complex side and simplify; never 'move terms across the equals sign'
Always look for a common factor first before attempting quadratic factorisation
In substitution, write brackets around every negative value: $(-3)^2 = 9$ , not $-3^2 = -9$
$(x + a)^2$ always has three terms — if your expansion has only two, you have missed the middle term $2ax$
Learn the difference-of-two-squares pattern $a^2 - b^2 = (a+b)(a-b)$ — it appears frequently on non-calculator papers

Frequently asked questions

Sources: AQA GCSE Mathematics (8300) Specification (2026 onwards); AQA 8300 Examiners' Reports 2022–2024 · Last reviewed May 2026 · AQA GCSE · Mathematics 8300 Higher

Notation, Vocabulary and Manipulation – Study Notes & Past Paper Style Questions | AQA GCSE Mathematics 8300 | Tutopiya

Notation, Vocabulary and Manipulation

At a glance

What you should be able to do

Study notes

Algebraic Notation

Key Vocabulary

Substitution

Collecting Like Terms

Expanding Brackets

Factorising

Rearranging Formulae

Quick recap

Exam tips

Frequently asked questions

What is the difference between an expression and an equation?

When can I collect like terms?

What does FOIL stand for and how do I use it?

How do I choose which factorising method to use?

What does 'change the subject' mean and how do I do it?

What is an identity and how do I prove one?