AQA GCSE Mathematics 8300: Notation, vocabulary and manipulation – Revision Notes

These notes condense section 3.2.1 Notation, vocabulary and manipulation (A1 to A7) of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). Content from the specification’s “Higher content only” column is marked Higher tier only; everything else can appear on Foundation and Higher papers. For full explanations and worked examples, use the study guide for this section.

Test yourself afterwards with the practice questions for this section. The topic-level Algebra revision notes cover graphs, equations and sequences. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free 10-minute diagnostics help you find gaps.

Tier map for 3.2.1

Ref	Both tiers	Higher tier only
A1	All notation	–
A2	All substitution	–
A3	Vocabulary, including identities	–
A4	Like terms, single brackets, common factors, index laws, surds, two binomials, x² + bx + c, difference of two squares	Two or more binomials, ax² + bx + c, algebraic fractions
A5	Standard formulae, changing the subject	–
A6	Equation vs identity, showing equivalence, algebraic arguments	Proof
A7	Expressions as functions (input → output)	Inverse and composite functions; f(x), fg(x), f⁻¹(x)

Paper 1 is non-calculator. Every skill in this section can appear there, so practise all of it by hand.

A1: Notation in one table

Long form	Write it as
a × b	ab
y + y + y, or 3 × y	3y
a × a, a × a × a	a², a³
a × a × b	a²b
a ÷ b	a/b
0.5x, 0.25y	½x (or x/2), ¼y (or y/4)
2 × (a + b)	2(a + b)

Answers are expected in simplest form without being told.

A2: Substitution in three steps

1. Write the expression again with each letter replaced by its value in brackets.
2. Powers first, then × and ÷, then + and −.
3. Give the units if the formula has them.

(−4)² = 16, but −4² = −16. Brackets decide which you mean.

Unfamiliar formulae are given in the question. The specification lists the kinematics formulae (v = u + at, s = ut + ½at², v² = u² + 2as) among those given in the exam. The quadratic formula, circle formulae and Pythagoras’ theorem are on the list you must learn.

A3: The key words

• Term: one part of an expression, such as −5x in 2x² − 5x.
• Expression: terms with no equals sign.
• Equation: true for particular values only.
• Formula: a rule connecting variables, such as v = d/t.
• Inequality: uses <, >, ≤ or ≥.
• Identity: true for all values; symbol ≡.
• Factor: divides exactly into the expression.

A4: Manipulation rules

Index laws

Rule	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	x⁴ × x⁵ = x⁹
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	x⁹ ÷ x² = x⁷
(aᵐ)ⁿ = aᵐⁿ	(x³)⁵ = x¹⁵
a⁰ = 1	7⁰ = 1
a⁻ⁿ = 1/aⁿ	x⁻² = 1/x²

Deal with the numbers first, then each letter. 6x⁵ × 4x² = 24x⁷: multiply 6 × 4, add the powers 5 + 2. Zero and negative indices are additional foundation content (N7); fractional indices are Higher tier only (N7).

Method: expanding two binomials

1. First × first, first × second, second × first, second × second.
2. Collect the two middle terms.
3. Check the sign of the last term: (+)(−) gives −.

(4x − 1)(x + 3) = 4x² + 12x − x − 3 = 4x² + 11x − 3

A squared bracket always has a middle term: (x + k)² = x² + 2kx + k².

Method: factorising x² + bx + c

1. List factor pairs of c.
2. Pick the pair that adds to b.
3. Write (x + p)(x + q) and expand to check.

Sign pattern	Brackets
+b, +c	both +
−b, +c	both −
c negative	one +, one −

Difference of two squares: a² − b² = (a + b)(a − b). It has no middle term, and there is no “sum of two squares” rule.

Higher tier only: factorising ax² + bx + c

1. Work out a × c.
2. Find two numbers with product ac and sum b.
3. Split bx into those two terms.
4. Factorise each pair; the bracket must repeat.

Higher tier only: products of three binomials

Expand any two brackets, simplify, then multiply each term of the result by the third bracket. Expect up to six terms before collecting.

Higher tier only: algebraic fractions

Task	Method
Simplify	Factorise top and bottom fully, cancel common brackets
Add or subtract	Common denominator; multiply out the numerators; simplify
Multiply	Factorise, cancel, then multiply across
Divide	Flip the second fraction, then multiply

Cancel factors (whole brackets), never single terms.

Surds inside brackets

Expand as normal and use √k × √k = k. Collect whole numbers together and surd terms together.

A5: Changing the subject

1. Clear fractions by multiplying both sides.
2. Expand any brackets.
3. Collect terms containing the new subject on one side, everything else on the other.
4. If the subject appears in two terms, factorise it out.
5. Divide, then take a root if the subject was squared.

The new subject should end up alone on one side, with no copy of it on the other side.

A6: Equation, identity, argument, proof

	What it is	How you show it
Equation	True for some values	Solve it
Identity	True for all values	Simplify one side until it matches the other
Argument	Algebra backs up a claim	Write the general case and simplify
Proof (Higher tier only)	Shows a claim holds in every case	General terms, full algebra, concluding sentence

Building blocks for proof: any integer n; even 2n; odd 2n + 1; consecutive integers n, n + 1; consecutive even numbers 2n, 2n + 2; consecutive odd numbers 2n + 1, 2n + 3. A multiple of k is written k × (an integer expression).

A7: Functions

Both tiers: a function machine turns an input into an output. Reverse a machine by doing the inverse operations in reverse order.

Higher tier only

Notation	Meaning	How to find it
f(3)	Output when the input is 3	Substitute 3 for x
f⁻¹(x)	Inverse function: undoes f	Write y = f(x), make x the subject, swap back to x
fg(x)	Composite: g first, then f	Put g(x) into f in place of x
gf(x)	f first, then g	Put f(x) into g

fg(x) and gf(x) are usually different.

Must-know distinctions

• Equation vs identity: = for some values, ≡ for all values.
• Expression vs equation: no equals sign vs an equals sign.
• Expand vs factorise: remove brackets vs put them in.
• (ab)² vs ab²: a²b² vs a × b × b.
• fg(x) vs gf(x): the inner function acts first.
• f⁻¹(x) vs 1/f(x): the inverse function is not the reciprocal.

Quick self-test

1. Write k × k × k × m ÷ 3 in simplest form.
2. Find the value of 3a² − ab when a = −2 and b = 5.
3. Name each: (i) 5t − 2, (ii) 5t − 2 = 13, (iii) 2(t + 1) ≡ 2t + 2, (iv) v = d/t.
4. Simplify (3m²n)³.
5. Expand and simplify (x − 5)(x + 8).
6. Factorise x² + 9x + 20.
7. Factorise p² − 64.
8. Make u the subject of v² = u² + 2as.
9. Show that (x + 4)² − 8x ≡ x² + 16.
10. (Higher) Factorise 2x² − 7x + 3.
11. (Higher) Simplify (x² + 5x)/(x² − 25).
12. (Higher) f(x) = 2x + 7 and g(x) = x². Find f⁻¹(x), gf(1) and fg(x).

Answers

1. k³m/3
2. 3(−2)² − (−2)(5) = 12 + 10 = 22
3. (i) expression, (ii) equation, (iii) identity, (iv) formula
4. 27m⁶n³ (3³ = 27; powers 2 × 3 = 6 and 1 × 3 = 3)
5. x² + 8x − 5x − 40 = x² + 3x − 40
6. 4 × 5 = 20 and 4 + 5 = 9, so (x + 4)(x + 5)
7. (p + 8)(p − 8)
8. u² = v² − 2as, so u = ±√(v² − 2as)
9. (x + 4)² = x² + 8x + 16; subtract 8x to get x² + 16 ✓
10. ac = 6; −6 and −1 have product 6 and sum −7. 2x² − 6x − x + 3 = 2x(x − 3) − 1(x − 3) = (2x − 1)(x − 3)
11. x(x + 5)/((x + 5)(x − 5)) = x/(x − 5)
12. f⁻¹(x) = (x − 7)/2; f(1) = 9, so gf(1) = 81; fg(x) = 2x² + 7

Where marks are usually lost

• Leaving an answer like 3 × a × b or 0.5x when the specification expects simplest form, ab-style notation and fraction coefficients.
• Substituting a negative without brackets, so a squared negative comes out negative.
• Multiplying the numbers but forgetting to apply the power to the coefficient: (3m²n)³ written as 3m⁶n³.
• Expanding a squared bracket as two squares with no middle term.
• Factorising only partly: 10p²q − 15pq² written as 5p(2pq − 3q²) instead of 5pq(2p − 3q).
• Writing (x − 4)(x − 5) for an expression whose last term is negative; check the signs by expanding.
• In a rearrangement, dividing only one term by the coefficient instead of the whole side.
• Cancelling x from x/(x − 5) to get 1/(−5): only whole factors cancel (Higher).
• Working out fg(x) by putting g into f the wrong way round (Higher).
• Giving numerical examples instead of a general argument on a “prove” question (Higher).

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.2.1 Notation, vocabulary and manipulation (A1 to A7) and Appendix: mathematical formulae.